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Geometry
3D Geometry
2015-10-28
www.njctl.org
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Table of Contents
Intro to 3-D SolidsViews & Drawings of 3-D SolidsSurface Area of a Prism
Surface Area of a CylinderSurface Area of a PyramidSurface Area of a Cone
Click on the topic to go to that section
Volume of a PrismVolume of a Cylinder
Volume of a PyramidVolume of a ConeSurface Area & Volume of SpheresCavaleri's PrincipleSimilar SolidsPARCC Sample Questions
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Throughout this unit, the Standards for Mathematical Practice are used.
MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.
Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.
If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Intro to 3-Dimensional Solids
Return to Table of Contents
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2-dimensional drawings use only the x and y axes
X
Y
Length
widthY
X
Length width
Y
X Length
width
Intro to 3-D Solids
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Y
X
Z
height
height
Y
X
3-dimensional drawings include the x, y and z-axis.
The z-axis is the third dimension.
The third dimension is the height of the figure
Intro to 3-D Solids
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Y
X
Z
height
height
YX
x
Y
Intro to 3-D Solids
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Y
X
Z
height
Y
X
X
Y
r
Intro to 3-D Solids
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To give a figure more of a 3-dimensional look, lines that are not visible from the angle the figure is being viewed are drawn as dashed line segments. These are called hidden lines.
Y
X
Z
height
height
Intro to 3-D Solids
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A Polyhedron (pl. Polyhedra) is a solid that is bounded by polygons, called faces. An edge is the line segment formed by the intersection of 2 faces. A vertex is a point where 3 or more edges meet
Face
Edge Vertex
Intro to 3-D SolidsSlide 12 / 311
The 3-Dimensional Figures discussed in this unit are:
Pyramids
CylindersPrisms
Intro to 3-D Solids
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The 3-Dimensional Figures discussed in this unit are:
. C
Cones: Spheres:
Intro to 3-D Solids
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Right Vs. Oblique
Right Right
In Right Prisms & Cylinders, the bases are aligned directly above one another. The edges are perpendicular with both bases.
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Right Vs. ObliqueIn Oblique Prisms & Cylinders, the bases are not aligned directly above one another. The edges are not perpendicular with the bases.
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Right Vs. Oblique
Right Oblique
Right Oblique
In Right Pyramids & Cones, the vertex is aligned directly above the center of the base.
In Oblique Pyramids & Cones, the vertex is not aligned
directly above the center of the base.
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Prisms have 2 congruent polygonal bases. The sides of a base are called base edges.
The segments connecting corresponding vertices are lateral edges.A
B
C
X Y
ZIn this diagram:There are 6 vertices: A, B, C, X, Y, & ZThere are 2 bases: ABC & XYZ.There are 6 base edges: AB, BC, AC, XY, YZ, & XZ.There are 3 lateral edges: AX, BY, & CZ.This prism has a total of 9 edges.
Intro to 3-D Solids
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The polygons that make up the surface of the figure are called faces. The bases are a type of face and are parallel and congruent to each other. The lateral edges are the sides of the lateral faces.
AB
C
X Y
Z
In this diagram:There are 2 bases: ABC & XYZ.
There are 3 lateral faces: AXBY, BYCZ, & CZAX.
This prism has a total of 5 faces.
Intro to 3-D Solids
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1
A AB
B DE
C FS
D CP
E FA
F CD
G NP
H BC
I DQ
AB C
DEF
M
N PQ
RS
Choose all of the base edges.
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2
A AB
B CD
C ER
D BN
E DQ
F QR
G MS
H AM
I CP
Choose all of the lateral edges.
AB C
DEF
M
N PQ
RS
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3 Chooses all of the bases.
A AFSM
B FERS
C EDQR
D ABCDEF
E CDQP
F BCPN
G MNPQRS
H ABNM
AB C
DEF
M
N PQ
RS
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4 Chooses all of the lateral faces.
A AFSM
B FERS
C EDQR
D ABCDEF
E CDQP
F BCPN
G MNPQRS
H ABNM
AB C
DEF
M
N PQ
RS
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5 Chooses all of the faces.
A AFSM
B FERS
C EDQR
D ABCDEF
E CDQP
F BCPN
G MNPQRS
H ABNM
AB C
DEF
M
N PQ
RS
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A pyramid has 1 base with vertices and the lateral edges go to a single vertex.
A
M
N P
RS
Q
This pyramid has: 6 lateral edges, 6 base edges, 12 edges (total) 7 vertices
Intro to 3-D Solids
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A pyramid has faces that are polygons: 1 base and triangles that are the lateral faces.
A
M
N PQ
RS
This pyramid has: 6 lateral faces, 1 base, 7 faces (total)
Intro to 3-D Solids
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6 Choose all of the base edges.
A VN
B KN
C VL
D LM
E VM
F VK
K
L
MN
V
G KL
H NM
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7 Choose all of the lateral edges.
A VN
B KN
C VL
D LM
E VM
F VK
G KL
H NM
K
L
MN
V
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8 How many edges does the pyramid have?
K
L
MN
V
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9 Choose all of the lateral faces.
A KNV
B NMV
C KLMN
D VML
E KLV
K
L
MN
V
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10 Choose all of the bases.
A KNV
B NMV
C KLMN
D VML
E KLV
K
L
MN
V
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11 How many faces does the pyramid have?
K
L
MN
V
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.
.
A
B
A cylinder has 2 bases which are congruent circles. The lateral face is a rectangle wrapped around the circles.
A & Bare the bases
of the cylinder.
Intro to 3-D Solids
A cylinder can also be formed by rotating a rectangle about an axis.
Click for sample animation
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A cone, like a pyramid, has one base which is a circle.
. N
V
N is thebase of the cone.
V is the vertex of the cone.
Intro to 3-D Solids
A cone can also be formed by rotating a right triangle about one of its legs.
Click for sample animation
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A sphere is a 3-dimensional circle in that every point on the sphere is the same distance from the center.
. C
Similar to a circle, a sphere is named by its center point. Sphere C is the solid shown above.
Intro to 3-D Solids
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12 Which solids have 2 bases?
A Prism
B Pyramid
C Cylinder
D Cone
E Sphere
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13 Which solid has one vertex?
A Prism
B Pyramid
C Cylinder
D Cone
E Sphere
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14 Which solid has more base edges than lateral edges?
A Prism
B Pyramid
C Cylinder
D Cone
E Sphere
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15 Which solid(s) have no vertices?
A Prism
B Pyramid
C Cylinder
D Cone
E Sphere
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16 Which solid is formed when rotating an isosceles triangle about its altitude?
A a prism
B a cylinder
C a pyramid
D a cone
E a sphere
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Euler's Theorem states that the number of faces (F), vertices (V), and edges (E) satisfy the formula F + V = E + 2
AB
C
X Y
Z
A
MN P
QRS
F = 5V = 6E = 9
5 + 6 = 9 + 211 = 11
F = 7V = 7
E = 127 + 7 = 12 + 2
14 = 14
Intro to 3-D Solids
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Example:A solid has 12 faces, 2 decagons and 10 trapezoids. How many vertices does the solid have?
V + F = E + 2V + 12 = 30 + 2V + 12 = 32V = 20
On their own, the 2 decagons & 10 trapezoids have2(10) + 10(4) = 60 edges. In a 3-D solid, each side is shared by 2 polygons. Therefore, the number of edges in the solid is 60/2 = 30.
Intro to 3-D Solids
click
click
click
click
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Example:A solid has 9 faces, 1 octagon and 8 triangles. How many vertices does the solid have?
V + F = E + 2V + 9 = 16 + 2V + 9 = 18V = 9
What information do you have? 9 faces & the 2 types of faces
Intro to 3-D Solids
click
click
click
click
What is the problem asking? Create an equation to represent the problem.
How are the number of edges in the 2-D faces, related to the number of edges in the polyhedron? Write a number sentence to describe this situation.
(1(8) + 8(3))/2(8 + 24)/2
32/216 edges
click
click
click
click
click
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17 A solid has 10 faces, one of them being a nonagon and 9 triangles. How many vertices does it have?
A 8
B 9
C 10
D 18
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18 A solid has 12 faces, all of them being pentagons. How many vertices does it have?
A 30
B 20
C 15
D 10
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19 A solid has 8 faces, all of them being triangles. How many vertices does it have?
A 24
B 12
C 8
D 6
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A cross-section is the locus of points of the intersection of a plane and a 3-D solid.
Cross-SectionIntro to 3-D Solids
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Think about it as if the plane were a knife and you were cutting the shape, what would the cut look like?
Cross-Section
Circle Ellipse
Parabola (with the inner section shaded)
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Cross-sections of a surface are a 2-dimensional figure.
Cross-sections of a solid are a 2-dimensional figure and its interior.
The top can be removed to see the cross section. (Try it out)
Cross-Section
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20 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides?
A square
B rectangle
C trapezoid
D hexagon
E rhombus
F parallelogram
G triangle
H circle
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21 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane perpendicular to the base and parallel to the non-intersecting sides?
A 72 sq inches
B 144 sq inches
C 187.06 sq inches
D 203.65 sq inches 12 in.
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22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base?
A square
B rectangle
C trapezoid
D hexagon
E rhombus
F parallelogram
G triangle
H circle
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23 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane that contains the diagonal of the base and is perpendicular to the base?
A 72 sq inches
B 144 sq inches
C 187.06 sq inches
D 203.65 sq inches
12 in.
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24 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base?
A square
B rectangle
C trapezoid
D hexagon
E rhombus
F parallelogram
G triangle
H circle
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25 What is the locus of points of a cube and a plane that intersects all of the faces?
A square
B rectangle
C trapezoid
D hexagon
E rhombus
F parallelogram
G triangle
H circle
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Views & Drawings of 3-D Solids
Return to Table of Contents
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Isometric drawings are drawings that look 3-D & are created on a grid of dots using 3 axes that intersect to form 120° & 60° angles.
Views & Drawings
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Example: Create an Isometric drawing of a cube.
Views & Drawings
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An Orthographic projection is a 2-D drawing that shows the different viewpoints of an object, usually from the front, top & side. Each drawing depends on your position relative to the figure.
Front Side
Top (from front)
Views & Drawings
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Consider these three people viewing a pyramid:
Views & Drawings
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Consider these three people viewing a pyramid:
The orange person is standing in front of a face, so their view is a triangle.
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Consider these three people viewing a pyramid:
The green person is standing in front of a lateral edge, so from their view they can see 2 faces.
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Consider these three people viewing a pyramid:
The purple person is flying over and can see the four lateral faces.
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26 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (front)
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27 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (top)
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28 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (top)
right square prism
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29 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a trapezoid
A (front)
right square prism
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30 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (front)
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31 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (above)
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32 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (above)
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33 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A (front)
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34 Given the surface shown, what would be the view from point A?
A a Rectangle
B a Square
C a Circle
D a Pentagon
E a Triangle
F a Parallelogram
G a Hexagon
H a Trapezoid
A
sphere
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AB
C (Looking down from above)What would the view be like from each position?
Views & Drawings
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A
What would the view be like from each position?
From A, how many columns of blocks are visible? - 3 columns How tall is each column? - first one is 4 high - second & third columns are each 2 blocks high
Click to reveal
Click to reveal
Views & Drawings
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B
What would the view be like from each position?
From B, how many columns of blocks are visible? - 2 columns
How tall is each column? - left one is 3 high - right one is 4 high
Click to reveal
Click to reveal
Views & Drawings
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C (Looking down from above)
What would the view be like from each position?
From C, how many columns of blocks are visible? - 3 columns How tall is each column? - all of them are 2 blocks high
Click to reveal
Click to reveal
Views & Drawings
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FrontSide
AboveDraw the 3 views.
Side ViewTop View
Front View
Move for Answer
Views & Drawings
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FrontSide
AboveDraw the 3 views.
AboveFront Side
Views & Drawings
Move for Answer
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Here are 3 views of a solid, draw a 3-dimensional representation.
Top FrontSide
L R
F
Views & Drawings
Move for Answer
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Here are 3 views of a solid, draw a 3-dimensional representation.
TopF
L R
Side Front
Views & Drawings
Move for Answer
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Surface Area of a Prism
Return to Table of Contents
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A Net is a 2-dimensional shape that folds into a 3-dimensional figure.
The Net shows all of the faces of the surface.
Net
6
646 4
12
4
Shown is the net of a right rectangular prism.
12
64
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The net shown is a right triangular prism. The lateral faces are rectangles. The bases are on opposite sides of the rectangles, although they do not need to be on the same rectangle.
Net
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The nets shown are for the same right triangular prism. Net
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Nets of oblique prisms have parallelograms as lateral faces.
Nets
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Rectangular Prisms
cube
ww w
H
HH
ℓ ℓ ℓ
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Base
Base height
Base
height
Base
A prism has 2 bases.
The base of a rectangular prism is a rectangle.
The height of the prism is the length between the two bases.
Rectangular Prisms
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The Surface Area of a figure is the total amount of area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift).
Area
Area
Area
Area
AreaArea
Top Area
Side
AreaFront Area Bottom
Area
Back Area
Side
Area
The Surface Area of a figure is the sum of the areas of each side of the figure.
Rectangular PrismsSlide 88 / 311
Finding the Surface Area of a Rectangular Prism
H
wℓ
Area of the Top = ℓ x w
Area of the Bottom = ℓ x w
Area of the Front = ℓ x H
Area of the Back = ℓ x H
Area of Left Side = w x H
Area of Right Side = w x H
The Surface Area is the sum of all the areas
S.A. = ℓw + ℓw + ℓH + ℓH + wH + wH
S.A. = 2 ℓw + 2 ℓH + 2wH
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Example: Find the surface area of the prism
74
3
Area of Top & Bottom Area of Right & Left
Area of Front & Back
A = 7(4) = 28u2A = 3(4) = 12 u2
A = 3(7) = 21 u2
Click
Click
Click
Total Surface Area = 2(28) + 2(12) + 2(21) = 56 + 24 + 42 = 122 units2
Click
Click
Finding the Surface Area of a Rectangular Prism
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35 What is the total surface area, in square units?
4
5
9
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36 What is the total surface area, in square units?
8
8
8
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37 Troy wants to build a cube out of straws. The cube is to have a total surface area of 96 in2, what is the total length of the straws, in inches?
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S.A. = 2B + PH
The Surface Area is the sum of the areas of the 2 Bases plus the Lateral Area (Perimeter of the base, P, times the height of the prism, H)
The Lateral Area is the area of the Lateral Surface. The Lateral Surface is the part that wraps around the middle of the figure (in between the two bases).
Another Way of Looking at Surface Area
Lateral Surface
Base
Base
Base
Base
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Base
Base
w
H
ℓ
Another formula for Surface Area of a right prism: S.A. = 2B + PH
B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism
S.A. = 2B + PH
S.A. = 2 ℓw + (2 ℓ +2w)H
S.A. = 2 ℓw + 2 ℓH + 2wH
Rectangular Prisms
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Base
Base
w
H
ℓ
In the surface area formula, 2B is the sum of the area of the 2 bases.
What does PH represent? The area of lateral faces or Lateral AreaClick
Rectangular PrismsAnother formula for Surface Area of a right prism:
S.A. = 2B + PH
B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism
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38 If the base of the prism is 12 by 6, what is the lateral area, in sq ft?
12 ft6 ft
4 ft
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39 The surface area of the rectangular prism is :
A 24 sq ft
B 144 sq ft
C 288 sq ft
D 48 sq ft
E 72 sq ft
12 ft6 ft
4 ft
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40 If 7 by 6 is base of the prism, what is the lateral area, in sq units?
7
96
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41 What is the total square units of the surface area?
7
96
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42 Find the value of y, if the lateral area is 144 sq units, and y by 6 is the base.
y
6 8
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43 What is the value of the missing variable if the surface area is 350 sq. ft.
A 7 ft
B 8.3 ft
C 12 ft
D 15 ft
x ft5 ft
10 ft
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44 Sharon was invited to Maria's birthday party. For a present, she purchased an iHome (a clock radio for an iPod or iPhone) which is contained in a box that measures 7 inches in length, 5 inches in width, and 4 inches in height. How much wrapping paper does Sharon need to wrap Maria's present?
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Other Prisms
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base
base height
base
base height
base
base
height
basebase height
A Prism has 2 Bases
The Base of a Prism matches the first word in the name of the prism. e.g. the Base of a Triangular Prism is a Triangle
The Height of the Prism is the length between the two bases
Other Prisms
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The Surface Area of a figure is the total amount of Area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift).
The Surface Area of a figure is the sum of the areas of each side of the figure
Area AreaArea
Area
Area
Area AreaArea
Area Area
Other Prisms
Triangular PrismNet of the Triangular Prism
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Finding the Surface Area of a Right PrismSurface Area: S.A. = 2B + PH B = Area of the triangular base = ½bh P = Perimeter of the triangular base = a + b + c H = Height of the prism
Lateral Area = PH = (a + b + c)H
The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases.
base
basePrism's
height
a
b
cH
P = a + b + c
ac
bc a
Lateral SurfaceHh
bB = ½ bh
Note: The formula above will work for any right prism.
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Example: Find the lateral area and surface area of the right triangular prism.
10
611
Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem. 62 + b2 = 102
36 + b2 = 100 b2 = 64 b = 8 units
Next, calculate the perimeter of your base. P = 6 + 8 + 10 = 24 unitsUse this to find the Lateral Area LA = PH = 24(11) = 264 units2
Other Prisms
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10
611
Example: Find the lateral area and surface area of the right triangular prism.
Then, calculate the area of your base, B B = (1/2)(8)(6) = 24 units2
Finally, calculate your Surface Area. SA = 2B + PH SA = 2(24) + (24)(11) SA = 48 + 264 = 312 units2
Other Prisms
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Example: Find the lateral area and surface area of the triangular prism. 99
9
12
Since it has a base that is an equilateral triangle, we need to find the height of the triangle using Pythagorean Theorem or the 30-60-90 Triangle Theorem. 4.52 + b2 = 92
20.25 + b2 = 81 b2 = 60.75 b = 4.5√3 units = 7.79 units
Next, calculate the perimeter of your base. P = 9 + 9 + 9 = 27 unitsUse this to find the Lateral Area LA = PH = 27(12) = 324 units2
Other PrismsSlide 110 / 311
Example: Find the lateral area and surface area of the triangular prism.
Then, calculate the area of your base, B B = (1/2)(9)(4.5√3) = 20.25√3 units2 = 35.07 units2
Finally, calculate your Surface Area. SA = 2B + PH SA = 2(35.07) + (27)(12) SA = 70.14 + 324 = 394.14 units2
Other Prisms
99
9
12
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45 The height of the triangular prism below is 11 ft, the height of the base is 3 ft, and the triangular base is an isosceles triangle. Find the surface area.
A 88 sq ft
B 132 sq ft
C 198 sq ft
D 222 sq ft 3 ft5 ft
11 ft
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46 The height of the triangular prism below is 3, and the triangular base is an equilateral triangle. Find the surface area.
A 64 sq ft
B 127.43 sq ft
C 72 sq ft
D 55.43 sq ft 8 ft3 ft
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47 Find the lateral area of the right prism.
5
56
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Finding the Surface Area of a Right PrismSurface Area : S.A. = 2B + PH B = Area of the regular hexagonal base = ½aP - a is the apothem of the regular base P = Perimeter of the base = b + c + d + e + f + g H = Height of the prism = HLateral Area = PH = (b + c + d + e + f + g)H
a
B = ½ aP
g
cb
H
e
f cde
fb
d
P = b + c + d + e + f + g
base
base
Prism's height
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a
B = ½ aP
Finding the Surface Area of a Right Prism
P = b + c + d + e + f + g
base
base
Prism's height
g
cb
H
e
f cde
fb
d
The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases.
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8 in
7 in
30°
4 in.
a
Example: Find the lateral area and surface area of the regular hexagonal prism.
Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle.
= 60° = central angle
= 30° = top angle of the triangle.
360 6
60 2
Click
Click
Click
Other Prisms
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Example: Find the lateral area and surface area of the regular hexagonal prism.
Next find the apothem using trigonometry, or special right triangles (if it applies). tan 30 =
atan30 = 4 tan30 tan30
4 a
a = 4√3 = 6.93 in.
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Other Prisms
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8 in
7 in
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B = (1/2)aP = (1/2)(4√3)(48) = 96√3 in2 = 166.28in2
Example: Find the lateral area and surface area of the regular hexagonal prism.
Next, calculate the perimeter of your base. P = 8(6) = 48 inUse this to find the Lateral Area LA = PH = 48(7) = 336 in2
Then, calculate the area of your base, B
Finally, calculate your Surface Area. SA = 2B + PH SA = 2(166.28) + (48)(7) SA = 332.56 + 336 = 668.56 in2
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8 in
7 in
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36°
3 in.
a
Example: Find the lateral area and surface area of the right prism.
The base is a regular pentagon.
6 ft
10 ft
Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle.
= 72° = central angle
= 36° = top angle of the triangle.
360 5
72 2
Other Prisms
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Example: Find the lateral area and surface area of the right prism.
The base is a regular pentagon.
Next find the apothem using trigonometry, or special right triangles (if it applies).
tan 36 = 3 a
atan36 = 3 tan36 tan36
a = 4.13 in.
6 ft
10 ft
Other Prisms
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Example: Find the lateral area and surface area of the right prism.
The base is a regular pentagon.
Next, calculate the perimeter of your base. P = 5(6) = 30 inUse this to find the Lateral Area LA = PH = 30(10) = 300 in2
Then, calculate the area of your base, B B = (1/2)aP = (1/2)(4.13)(30) = 61.95 in2
Finally, calculate your Surface Area. SA = 2B + PH SA = 2(61.95) + (30)(10) SA = 123.9 + 300 = 423.9 in2
6 ft
10 ft
Other Prisms
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Example: Find the lateral area and surface area of the right prism.
8 3
7
6
5Angles are right angles.
First, calculate the perimeter of your base. P = 8 + 7 + 5 + 4 + 3 + 3 P = 30 unitsUse this to find the Lateral Area LA = PH = 30(6) = 180 units2
Other Prisms
Then, calculate the area of your base, B B = 7(5)+3(3) = 44 units2
Finally, calculate your Surface Area. SA = 2B + PH SA = 2(44) + (30)(6) SA = 88 + 180 = 268 units2
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48 Find the lateral area of the right prism.
8
11
The base is a regular hexagon.
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49 Find the total surface area of the right prism.
The base is a regular hexagon.
8
11
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50 Find the total surface area of the right prism.
4
4 3
2
10
9
All angles are right angles.
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y
56
51 The right triangular prism has a surface area of 150 sq ft. Find the height of the prism.
A 5 ftB 6 ftC 7.81 ftD 6.38 ft
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Surface Area of a Cylinder
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height
radius
base
base
height
radi
us
base base
Cylinders
A Cylinder is a solid w/ 2 circular bases that lie in || planes. Because each base is a circle, it contains a radius. The remaining measurement that connects the 2 bases is the height of the cylinder.
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8
radius
The net of a right cylinder is two circles and a rectangle that forms the lateral surface.
8
x
What is the length of x? - The circumference of the circle (base)
radius
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Cylinders
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Base
Base
height
Base
height Lateral Surface
Base
Finding the Surface Area of a Right Cylinder
Surface Area : S.A. = 2B + PH B = Area of the circular base = πr2 C = Perimeter of the Circular base (Circumference) = 2πr H = Height of the prism
Lateral Area = CH = 2πrH
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Base
Base
height
Base
height Lateral Surface
Base
Finding the Surface Area of a Right Cylinder
The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the cylinder between the circular bases.
Therefore, the Surface Area of a Cylinder can be simplified to the equation below. SA = 2πr2 + 2πrH
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8
r = 4
Example: Find the lateral area and surface area of the right cylinder.
LA = 2πrhLA = 2π(4)(8)LA = 64π units2
LA = 201.06 units2
SA = 2πr2 + 2πrhSA = 2π(4)2 + 2π(4)(8)SA = 32π + 64πSA = 96π units2
SA = 301.59 units2
Finding the Surface Area of a Right Cylinder
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34
d = 16
SA = 2πr2 + 2πrhSA = 2π(8)2 + 2π(8)(30)SA = 128π + 480πSA = 608π units2
SA = 1,910.09 units2
Example: Find the lateral area and surface area of the right cylinder.
LA = 2πrhLA = 2π(8)(30)LA = 480π units2
LA = 1507.96 units2
162 + h2 = 342
256 + h2 = 1156h2 = 900h = 30Note: 16-30-34 = 2(8-15-17) Pyth. Tripleclick
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Example: Find the lateral area and surface area of the right cylinder when the base circumference is 16π ft & the height is 10 ft.
SA = 2πr2 + 2πrhSA = 2π(8)2 + 2π(8)(10)SA = 128π + 160πSA = 288π ft2
SA = 904.78 ft2
LA = 2πrhLA = 2π(8)(10)LA = 160π ft2
LA = 502.64 ft2
C = 2πr16π = 2πr 2π 2π8 ft = r
Cylinders
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h = 12
r = 7
52 Find the lateral area of the right cylinder.
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h = 12
r = 7
53 Find the surface area of the right cylinder. Use 3.14 as your value of π & round to two decimal places.
A 1200 sq in.B 307.72 sq in.C 835.24 sq in.D 1670.48 sq in.
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54 Find the lateral area of the right cylinder.
13
r = 5
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h = 12
55 Find the lateral area of the right cylinder.
Base area is 36π units2
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h = 12
56 Find the surface area of the right cylinder.
Base area is 36π units2
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r = 8 in.
h
57 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder. Use 3.14 as your value of π.
A 7 in.B 8 in.C 5 in.D 6 in.
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58 A food company packages soup in aluminum cans that have a diameter of 2 1/2 inches and a height of 4 inches. Before shipping the cans off to the stores, they add their company label to the can which does not cover the top and bottom. If the company is shipping 200 cans of soup to one store, how much paper material is required to make the labels?
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59 Maria's mom baked a cake for her daughter's birthday party. The diameter of the cake is 9 inches and the height is 2 inches.
How much base frosting (pink in the picture below) was required to cover the cake?
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Surface Area of a Pyramid
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Slide 144 / 311
A Pyramid is a polyhedron in which the base is a polygon & the lateral faces are triangles with a common vertex.
Lateral Edges are the intersection of 2 lateral faces Vertex
LateralFace
LateralEdge
Base
Pyramids
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Net
This is a right square pyramid. Another name for it is pentahedron.Hedron is a suffix that means face. Why is this a pentahedron?
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Slant Height
ℓ
The Pyramid has a square base and 4 triangular facesThe triangular faces are all isosceles triangles if its a right pyramid.The Height of each triangular face is the Slant Height of the pyramid if it is a regular pyramid (labeled as , or a cursive lower case L).
Surface Area = Sum of the Areas of all the sides
ℓ
Heightof theTriangle
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Square Base (B)
Slant Height ( )ℓ
Pyramid's Height (h)
Segment Lengths in a Pyramid
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Example: Find the value of x.
a2 + 122 = 132
a2 + 144 = 169a2 = 25a = 5Note: 5-12-13 Right Triangle
Therefore x = 2(5) = 10x
1312
Segment Lengths in a Pyramid
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Example: Find the value of x.
Base Area of the right square pyramid is 64 u2.
x8
Square Base has an area of 64, so64 = y2
y = 8, so a = 4 of the right triangle.
42 + 82 = x2
16 + 64 = x2
x2 = 80x = 8.94 units
Segment Lengths in a Pyramid
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Example: Find the length of the slant height.
r
This is a regular hexagonal pyramid.
r = 6lateral edge = 12
Segment Lengths in a Pyramid
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First, find the height of the pyramid using Pythagorean Theorem.
h12
6
62 + h2 = 12236 + h2 = 144h2 = 108h = 6√3 = 10.39
Note: 30-60-90 triangler
Segment Lengths in a Pyramid
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Second, find the apothem of the hexagonal base.
a6 6
3 3
32 + a2 = 62
9 + a2 = 36a2 = 27a = 3√3 = 5.20Note: 30-60-90 triangle
= 60° = central
Note: equilateral
= 30° = top of the .
360 6
60 2r
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(3√3)2 + (6√3)2 = 2
27 + 108 = 2
2 = 135 = 3√15 = 11.62
ℓℓ
ℓℓ
Last, find the slant height of your pyramid w/ the apothem & height.
a = 3√3
ℓ h = 6√3
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Segment Lengths in a Pyramid
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60 Find the value of the variable.
16
x6
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61 Find the value of the variable.
12
11x
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62 Find the value of the variable.
x6
area of the base is 36 u2
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63 Find the value of the slant height.
r
r = 8
lateral edge = 17
Regular Hexagonal Pyramid
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64 Find the value of the slant height.
a
a = 9
lateral edge = 12
Regular Hexagonal Pyramid
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Square Base (B)
Slant Height ( )
Pyramid's Height (h)
ℓ
Surface Area = B + ½P and Lateral Area = ½P = Slant HeightP = Perimeter of BaseB = Area of Base
Surface Area of a Regular Pyramid
ℓ ℓℓ
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Square Base (B)
Slant Height ( )
Pyramid's Height (h)
Why is the Surface Area SA = B + P ? 1 2
Surface Area is the sum of all of the areas that make up the solid. In our diagram, these are 4 triangles & 1 square.Asquare = s s = s2 = B
A∆ = s 1 2 ℓ
ℓ
ℓ
Surface Area of a Regular Pyramid
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Why is the Surface Area SA = B + P ? 1 2
Since there are 4 ∆s, we can multiply the area of each ∆ by 4. Therefore, our Surface Area for the Pyramid above isSA = s2 + 4(1/2)sSA = s2 + (1/2)(4s)SA = B + 1/2 P
s
ℓ
Net of Pyramid
ℓℓ
ℓ
Surface Area of a Regular Pyramid
ℓ
Slide 162 / 311
ℓ = 7
s = 6
Example: Find the lateral area and the surface area of the pyramid.
LA = 1/2 P ℓLA = 1/2 (24)(7)LA = 12(7)LA = 84 units2
SA = B + 1/2 P ℓSA = 62 + 1/2 (24)(7)SA = 36 + 84SA = 120 units2
Surface Area of a Regular Pyramid
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Example: Find the lateral area and the surface area of the pyramid.
First, calculate the slant height.32 + 82 = ℓ 29 + 64 = ℓ2
73 = ℓ2 ℓ = 8.54Next, calculate the LA & SA
LA = 1/2 P ℓLA = 1/2 (24)(8.54)LA = 12(8.54)LA = 102.48 units2
SA = B + 1/2 P ℓSA = 62 + 1/2 (24)(8.54)SA = 36 + 102.48SA = 138.48 units2
h = 8
s = 6
Surface Area of a Regular Pyramid
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Example: Find the lateral area and the surface area of the pyramid.
10
8
ℓ
First, calculate the slant height.
82 + ℓ 2 = 102
64 + ℓ 2 = 100 ℓ 2 = 36 ℓ = 6
s = 16
e = 10
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Surface Area of a Regular Pyramid
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Example: Find the lateral area and the surface area of the pyramid.
LA = 1/2 P ℓLA = 1/2 (64)(6)LA = 32(6)LA = 192 units2
SA = B + 1/2 P ℓ SA = 162 + 1/2 (64)(6)SA = 256 + 192SA = 448 units2
Next, calculate the LA & SA
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s = 16
e = 10
Surface Area of a Regular Pyramid
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Example: Find the lateral area and the surface area of the pyramid.
a
a = 4lateral edge = 8
Regular Pentagonal Pyramid
Surface Area of a Regular Pyramid
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72°
36°36°4
x
r
Example: Find the lateral area and the surface area of the pyramid.First, find the radius & side length of the regular pentagon using the apothem & trigonometric ratios
tan36 =
x = 4tan36 = 2.91
Therefore, s = 2(2.91) = 5.82
= 36° = top of the .
360 5
72 2
x 4
4 rcos36 =
rcos36 = 4 cos36 cos36
r = 4.94
= 72° = centralClick
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Surface Area of a Regular PyramidSlide 168 / 311
Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pythagorean Theorem.
8
2.91
2.912 + ℓ 2 = 82
8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45
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Surface Area of a Regular Pyramid
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Last, find the lateral area & surface area of the pyramid.
LA = 1/2 P ℓLA = 1/2 (29.1)(7.45)LA = 108.40 units2
SA = B + 1/2 P ℓSA = 1/2 (4)(29.1) + 1/2 (29.1)(7.45)SA = 58.2 + 108.40SA = 166.6 units2 click
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Surface Area of a Regular Pyramid
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65 Find the lateral area of the right pyramid.
s = 10
ℓ = 9
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66 Find the surface area of the right pyramid.
s = 10
ℓ = 9
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67 Find the lateral area of the right pyramid.
base
e = 10
area = 16
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68 Find the surface area of the right pyramid.
base
e = 10
area = 16
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a
a = 5h = 12
Regular Octagonal Pyramid
69 Find the lateral area of the right pyramid.
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a
a = 5h = 12
Regular Octagonal Pyramid
70 Find the surface area of the right pyramid.
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Hint: The pyramid is NOT regular. So, B + 1/2 P ℓ doesn't work. Instead, draw a net of the pyramid & find each area.
71 Find the surface area of the right pyramid.
30
12
8
Hint
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Surface Area of a Cone
Return to Table of Contents
Slide 178 / 311
r
height
Slant Height ℓ
Lateral SurfaceSlant Height ℓ
Base
The Base of the cone is a circle
The length of the circular portion of the Lateral Surface is the same as the Circumference of the Circlular Base.
The Slant Height is the length of the diagonal slant of the cone from the top to the edge of the base.
The Height of the cone is the length from the top to the center of the circular base.
Cones
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Surface Area = Area of the Base + Lateral AreaLateral Area= ½P ℓS.A. = B + ½P ℓ ℓ = Slant HeightP = Perimeter of Circular BaseB = Area of Circular BaseBecause the base is a circle. P = Circumference = 2πrL.A. = ½(2πr) ℓ = πr ℓ S.A. = πr2 + πr ℓ
Finding the Surface Area of a Right Cone
Lateral SurfaceSlant Height ℓ
Base
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LA = πr ℓ = π(6)(8)LA = 48π units2
LA = 150.80 units2
SA = πr2 + πr ℓ = π(6)2 + π(6)(8) = 36π + 48πSA = 84π units2
SA = 263.89 units2
Example: Find the lateral area and surface area of the right cone.
= 8
r = 6
ℓ
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Example: Find the lateral area and surface area of the right cone.
h = 8
C = 12π unitsC = 2πr12π = 2πr 2π 2π6 units = r
62 + 82 = ℓ2
36 + 64 = ℓ2
100 = ℓ2
10 units = ℓ
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Example: Find the lateral area and surface area of the right cone.
h = 8
C = 12π units
SA = πr2 + πr ℓ = π(6)2 + π(6)(10) = 36π + 60πSA = 96π units2
SA = 301.59 units2
LA = πr ℓ = π(6)(10)LA = 60π units2
LA = 188.50 units2
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72 Find the lateral area of the right cone, in square units.
r = 4
ℓ = 9
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r = 4
ℓ = 9
73 Find the surface area of the right cone, in square units.
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74 Find the lateral area of the right cone, in square units.
h = 9
Base Area = 16π units2
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75 Find the surface area of the right cone, in square units.
h = 9
Base Area = 16π units2
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76 Find the length of the radius of the right cone if the lateral area is 50π units2?
ℓ = 10
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ℓ = 10
77 Find the height of the right cone if the lateral area is 50π units2?
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78 Find the slant height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?
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79 Find the height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?
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80 The Department of Transportation keeps 4 piles of road salt for snowy days. Each conical shaped pile is 20 feet high and 30 feet across at the base. During the summer the piles are covered with tarps to prevent erosion. How much tarp is needed to cover the conical shaped piles so that no part of them are exposed?
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Volume of a Prism
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Slide 193 / 311
The volume of a solid is the amount of cubic units that a solid can hold.
Where area used square units, volume will use cubic units.
Prisms
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Base
height
Base
ℓw
HV = BH
Specific PrismsRectangular Prism: V = ℓwHCube: V = s3
Finding the Volume of a Prism
Prisms
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Does a prism need to be a right prism for the volume formula to work?
Think of a ream of paper
Stacked nicely it has 500 sheets.
If the stack is fanned, it still has 500 sheets.
So the volume doesn't change if the prism, stack of paper, is right or oblique.The formula V = BH works for all prisms.
Prisms
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Example: Find the volume of the rectangular prism with a length of 2, a width of 6, and a height of 5.
V = ℓ w HV = 2(6)(5)V = 60 units3
Prisms
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Example: The volume of a box is 48 ft3. If the height is 4 ft and width is 6 ft, what is the length?
V = ℓ w H48 = ℓ(6)(4)48 = 24 ℓ 24 242 ft = ℓ
PrismsSlide 198 / 311
Example: Find the volume of the prism shown below.
10
611
Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem. 62 + b2 = 102
36 + b2 = 100 b2 = 64 b = 8 units
Prisms
Next, calculate the area of your base, B B = (1/2)(8)(6) = 24 units2
Finally, calculate your Volume. V = BH V = 24(11) V = 264 units3
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Example: The volume of a cube is 64 m3, what is area of one face?
V = s3
64 = s3
4 m = s
Area of one faceA = 4(4)A = 16 m2
PrismsSlide 200 / 311
4 in
7 in
30°4 in.
x in.
Because the base is a regular polygon, we need to calculate the side length. To begin, figure out the central angle & top angle in the triangle.
= 60° = central angle
= 30° = top angle of the triangle.
360 6
60 2
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Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in.
Prisms
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4 in
7 in
Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in.
Then, calculate the side length of your base. s = 2(2.31) = 4.62 in
Next, find the value of x using trigonometry, or special right triangles (if it applies). tan 30 =
4tan30 = x x = 4√3 = 2.31 in. 3
x 4
Prisms
30°4 in.
x in.
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4 in
7 in
Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in.
Next, use your value of s to find the Perimeter of your base P = 6(4.62) = 27.72 in
Prisms
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Then, calculate the area of your base, B B = (1/2)aP = (1/2)(2.31)(27.72) = 32.02 in2
Finally, calculate your Volume. V = Bh V = 32.02(7) V = 224.14 in3
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81 What is the volume of a rectangular prism with edges of 4, 5, and 7?
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82 What is the volume of a cube with edges of 5 units?
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83 If the volume of a rectangular prism is 64 u3 and has height 8 and width 4, what is the length?
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84 If a cube has volume 27 u3, what is the cubes surface area?
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85 Find the volume of the prism.
15
1220
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86 Find the volume of the prism.
7
2
6
6
6
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87 Find the volume of the prism.
8
11
The base is a regular hexagon.
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88 A high school has a pool that is 25 yards in length, 60 feet in width, and contains the depth dimensions shown in the figure below.
If one cubic yard is about 201.974 gallons, how much water is required to fill the pool?
Shallow end
Deep end
3 ft9 ft
2 yds 4 yds19 yds
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Volume of a Cylinder
Return to Table of Contents
Slide 212 / 311
base
base
height
r
r
Finding the Volume of a Cylinder
V = BhV = πr2h
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Example: Find the volume of the cylinder with a radius of 4 and a height of 11.
V = π(4)2 (11)V = 176π units3
V = 552.92 units3
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Example: The surface area of a cylinder is 96π units2, and its radius is 4 units. What is the volume?
V = π(4)2 (8)V = 128π units3
V = 402.12 units3
SA = 2πr2 + 2πrh96π = 2π(4)2 + 2π(4)h96π = 32π + 8πh-32π -32π 64π = 8πh 8π 8πh = 8 units
Cylinders
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89 Find the volume of the cylinder with radius 6 and height 8.
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90 Find the volume of the cylinder with a circumference of 18π units and a height of 6.
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r = 8
h
91 Find the volume of the cylinder with a surface area of 653.12 u2 & a radius of 8 units. Use 3.14 as your value of π.
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92 The volume of a cylinder is 108π u3, and the height is 12 units. What is the surface area?
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93 The height of a cylinder doubles, what happens to the volume?
A Doubles
B Quadruples
C Depends on the cylinder
D Cannot be determined
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94 The radius of a cylinder doubles, what happens to the volume?
A Doubles
B Quadruples
C Depends on the cylinder
D Cannot be determined
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24"
4"
3"
95 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube?
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Volume of a Pyramid
Return to Table of Contents
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Finding the Volume of a Pyramid
V = 1/3 BhSquare Base (B)
Slant Height ( )
Pyramid's Height (h)
ℓ
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Example: Find the volume of the pyramid.
54
6
V = 1/3 BhB = 5(4) = 20h = 6 unitsV = 1/3 (20)(6)V = 40 units3
Volume of Pyramids
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Example: Find the volume of the pyramid.
88
5
88
5
4
h
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Volume of PyramidsSlide 226 / 311
96 Find the volume of the pyramid.
76
5
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97 Find the volume of the pyramid.
66
8
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98 Find the volume of the pyramid.
12
12
10
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Example: Find the volume of the pyramid.
a
a = 4lateral edge = 8
Regular Pentagonal PyramidFirst, find the side length of the regular pentagon using the apothem & trigonometric ratios.
= 72° = central
= 36° = top angle of the .72 2
360 5
tan36 =
x = 4tan36 = 2.91
Therefore, s = 2(2.91) = 5.82
x 4
Volume of Pyramids
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Example: Find the volume of the pyramid.Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pyth. Theorem.
8
2.91
ℓ
2.912 + ℓ 2 = 82
8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45
Then, use the slant height & apothem w/ Pyth. Theorem to find the height.
7.45
4
hClic
k
42 + h2 = 7.452
16 + h2 = 55.5319h2 = 39.5319h = 6.29
Volume of Pyramids
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a
a = 4lateral edge = 8
Regular Pentagonal Pyramid
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Example: Find the volume of the pyramid.
Last, find the Area of your Base & Volume.
B = 1/2 aPB = 1/2 (4)(29.1)B = 58.2 units2
V = 1/3 BhV = 1/3 (58.2)(6.29)V = 122.03 units3
Volume of Pyramids
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a
a = 4lateral edge = 8
Regular Pentagonal Pyramid
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99 Find the volume of the right pyramid.
a
a = 5h = 12
Regular Octagonal Pyramid
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100 Find the volume of the right pyramid.
8
11
The base is a regular hexagon.
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A truncated pyramid is a pyramid with its top cutoff parallel to its base.
Find the volume of the truncated pyramid shown.
22
66
9
3Vtruncated = Vbig - Vsmall
Bbig = 6(6) = 36hbig = 3 + 9 = 12Vbig = 1/3 (36)(12)Vbig = 144 units3
Bsmall = 2(2) = 4hsmall = 3Vsmall = 1/3 (4)(3)Vsmall = 4 units3
Vtruncated = 144 - 4 Vtruncated = 140 units3
Volume of Pyramids
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101 Find the volume of the truncated pyramid.
22
8
8
12
3
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102 The table shows the approximate measurements of the Red Pyramid in Egypt and the Great Pyramid of Cholula in Mexico.
Approximately, what is the difference between the volume of the Red Pyramid and the volume of the Great Pyramid of Cholula?
A 6,132,867 cubic meters
B 4,455,000 cubic meters
C 2,777,133 cubic meters
D 1,677,867 cubic meters
Length Width HeightRed Pyramid 220 m 220m 104 mGreat Pyramid of Cholula 450 m 450 m 66 m
Ans
wer
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103 Salt water comes in cylindrical containers that measure 10 feet high and have a diameter of 8 feet. Determine the height of the aquarium that should be used in the design. Show that your design will be able to store at least 3 cylindrical containers of water. When you finish, enter your value for h1 into your SMART Responder.
The Geometryville Aquarium is building a new tank space for coral reef fish shown in the figure below. The laws say that the dimensions of the tank must have a maximum length of 14 feet, a maximum width of 10 feet and a maximum height of 16 feet.
w
h1
h2
ℓ
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Volume of a Cone
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r
height
Slant Height ℓ
Finding the Volume of a Cone
V = 1/3 Bh
V = 1/3πr2 h
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Example: Find the volume of the cone.
9
7
V = 1/3 πr2 hV = 1/3 π(7)2 (9)V = 147π units3
V = 461.81 units3
Volume of a Cone
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Example: Find the volume of the cone.
12
4
V = 1/3 πr2 hV = 1/3 π(4)2 (8.94)V = 47.68π units3
V = 149.79 units3
r = 4, so d = 8With the right triangle, use Pythagorean Theorem to find the height of the pyramid.h2 + 82 = 122
h2 + 64 = 144h2 = 80, h = √80 = 8.94
Volume of a ConeSlide 242 / 311
Example: Find the volume of the cone, with lateral area of 15π units2 and a slant height 5 units.
LA = πr ℓ 15π = πr(5)15π = 5πr 5π 5π3 units = r
1) You know the Lateral area & slant height, so use the Lateral Area formula to calculate the radius.
Volume of a Cone
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h2 + 32 = 52
h2 + 9 = 25 h2 = 16 h = 4Note: 3-4-5 Pyth. Triple
2) Next, use the slant height & radius to calculate the height of the cone using Pythagorean Theorem.
Volume of a Cone
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V = 1/3 πr2 h
V = 1/3 π(3)2 (4)V = 12π units3
V = 37.70 units3
3) Last, calculate the volume of the cone.
Volume of a Cone
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104 What is the volume of the cone?
8
d = 10
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105 What is the volume of the cone?
r = 4
= 9ℓ
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106 What is the volume of the cone?
1040°
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107 What is the volume of the truncated cone?
r = 8
r = 4
6
6
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Surface Area & Volume of Spheres
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Recall the Definition of a Circle
The locus of points in a plane that are the same distance from a point called the center of the circle.
X
Y
Every point on the above circle is the same distance from the origin in the x, y plane.
Y
X
Spheres
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The locus of points in space that are the same distance from a point.
Y
X
Z
Every point on the sphere above on the left side, is the same distance from the origin in space, the x, y, z plane.
X
Y
Y
X
Spheres
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Y
X
Z
The Great Circle of a sphere is found at the intersection of a plane and a sphere when the plane contains the center of the sphere.
Spheres
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Y
X
ZGreat Circles
Each of these planes intersects the sphere, and the plane contains the center of the sphere
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InternationalDate Line
Great Circles The Earth has 2 Great Circles: Can you name them?
Click to reveal picture
The Equator The Prime Meridian w/ the International Date Line
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Great Circle
The Great Circle separates the Sphere into two equal halves at the center of the sphere.
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Each half is called a Hemisphere
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Cross Sections
A Cross Section is found by the intersection of a plane and a solid.
Cross - Section
(Click the top hemisphere to see the cross section.)
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.small circles great
circle
The farther the cross section of the sphere is taken from its center the smaller the circle.
Cross Sections
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82 8
r
Example: Find the radius of the cross section of the sphere that has a radius of 8 if the cross section is 2 from the center.
22 + r2 = 82
4 + r2 = 64r2 = 60r = √60 = 2√15 = 7.75
Cross Sections
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4
Example: A cross section of a sphere is 4 units from the center of the sphere and has an area of 16π units2. What is area of the great circle? Leave your answer in terms of π.
16π = πr2
r = 4 units in the cross section42 + 42 = r2 32 = r2
r =√32 = 4√2 = 2.83 = radius of sphereA = π(√32)2
A = 32π units2
Cross Sections
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108 What is the area of the cross section of a sphere that is 6 units from the center of the sphere if the sphere has radius 8 units?
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109 What is the area of the great circle if a cross section that is 3 from the center has a circumference of 10π?
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110 The circumference of the great circle of a sphere is 12π units and a cross section has a circumference of 8π units. How far is the cross section from the center?
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rS.A. = 4πr2
Finding the Surface Area of the Sphere
Why is there no formula for lateral area?
A sphere doesn't have any bases, so the lateral area is the same as the surface area.
Click to reveal
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r V = πr3 4 3
Finding the Volume of the Sphere
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Example: Find the surface area & volume of a sphere with radius of 6 ft.
SA = 4π(6)2
SA = 144π units2
SA = 452.39 units2
V = π(6)3
V = 288π units3 V = 904.78 units3
4 3
Finding the Volume of the Sphere
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Example: Find the surface area & volume of a sphere that a great circle with area 24π units2?
SA = 4π(4.9)2
SA = 96.04π units2
SA = 301.72 units2
V = π(4.9)3
V = 156.87π units3 V = 492.81 units3
4 3
24π = πr2
π πr2 = 24r = 4.90 units
Finding the Volume of the Sphere
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Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume of the sphere?
Radius of Cross Section36π = πr2
π πr2 = 36r = 6 units
Radius of Sphere102 + 62 = R2 136 = R2
R = √136 = 11.66 units
Finding the Volume of the Sphere
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SA = 4π(√136)2
SA = 544π units2
SA = 1,709.03 units2
V = π(√136)3
V = 2,114.69π units3 V = 6,643.50 units3
4 3
Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume of the sphere?
Finding the Volume of the Sphere
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111 Find the surface area of a sphere with radius 10.
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112 Find the volume of a sphere with radius 10.
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113 What is the surface area of a sphere if a cross section 7 units from the center has an area of 50.26 units2?
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114 What is the volume of a sphere if a cross section 7 units from the center has an area of 50.26 units2?
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115 The volume of a sphere is 24π units3. What is the area of a great circle of the sphere?
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116 A recipe calls for half of an orange. Shelly use an orange that has a diameter of 3 inches. She wraps the remaining half of orange in plastic wrap. What is the amount of area that Shelly has to cover?
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Cavalieri's Principle
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Cavalieri's Principle
If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume.
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1414 14
Which solid has the greatest volume?
224π703.72
None: All of the solids have the same volume.Click
Cavalieri's Principle
2π8
4π
44
224π703.72
224π703.72Click Click Click
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Example: A sphere is submerged in a cylinder. Both solids have a radius of 4. What is the volume of the cylinder not occupied by the sphere?
volume of cylinder - volume of sphere
Cavalieri's Principle
π(4)2 (8) - 4/3 π(4)3
128π - 256/3 π 128/3 π units3 Click
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The result shows that the left over volume is equal to what other solid?
cone
According to Cavalieri, what can be said about the cross section? The cross section of the great circle of the sphere is equal to the circle cross section of the cylinder. Click
Cavalieri's Principle
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Example:What is the radius of a sphere made from the cylinder of modeling clay shown?
If you are using clay to model both solids, what measurement is the same? Volume
15
5
Cavalieri's Principle
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Therefore, calculate the volume of the cylinder first.
Cavalieri's Principle
V = π(5)2 (15) V = 375π units3
Then create an equation to represent the problem and solve for r.
375π = 4/3 πr3
375 = 4/3 r3
281.25 = r3
r = 6.55 units
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15
5
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117 These 2 solids have the same volume, find the value of x.
11
r = 6
11
x 9
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118 These 2 solids have the same volume, find the value of x.
12
x
12
10
8
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Prism C
B = 20 in2
x
Prism D
B = 20 in2
y
Two prisms each with a base area of 20 square inches are shown.
Which statements about prisms C and D are true. Select all that apply. (Statements are on the next slide.)
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119 Two prisms each with a base area of 20 square inches are shown.Which statements about prisms C and D are true. Select all that apply.
A If x > y, the area of a vertical cross section of prism C is greater than the area of a vertical cross section of prism D.
B If x > y, the area of a vertical cross section of prism C is equal to the area of a vertical cross section of prism D.
C If x > y, the area of a vertical cross section of prism C is less than the area of a vertical cross section of prism D.
D If x = y, the volume of prism C is greater than the volume of prism D, because prism C is a right prism.
E If x = y, the volume of prism C is equal to the volume of prism D because the prisms have the same base area.
F If x = y, the volume of prism C is less than the volume of prism D because prism D is an oblique prism.
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Similar Solids
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Corresponding sides of similar figures are similar.
The prisms shown are similar. Find the values of x and y.
4
x2 6
9y
4 6 = 4
6=
Similar Solids
x 9
36 = 6x 6 6 6 = x
4y = 12 4 4 y = 3
2 y
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4
x2 6
9y
The ratio of similarity, k, is the common value that is multiplied to preimage to get to the image.
- Hint: it's the ratio of image : preimage
If the smaller prism is the preimage, then the value of k is
If the larger prism is the preimage, then the value of k is
click for the hint
Similar Solids
3/2
2/3
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120 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of x.
8
8
16
h2
x
y
3
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121 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of y.
8
8
16
h 2
x
y
3
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122 The pyramid on the left is the preimage and is similar to the image on the right. Find the value of h.
8
8
16
h 2
x
y
3
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4
62
6
93
Consider the example of the prisms from earlier. The ratio of similarity from the smaller solid to the larger is 2:3.
Calculate the surface area of both solids. How do they compare? SAsmall = 2(6)(2) + 16(4) = 88 units2 SAbig = 2(3)(9) + 24(6) = 198 units2 SA Similarity ratio = 88:198 = 4:9 = 22:32
How do their volumes compare? Vsmall = 2(4)(6) = 48 units3 Vbig = 6(3)(9) = 162 units3 V Similarity ratio = 48:162 = 8:27 = 23:33
Similar Solids
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Comparing Similar Figures
length in image
length in preimage= k
area in image
area in preimage = k2
volume in image
volume in preimage = k3
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How many times bigger is the surface area of the sphere to the right?
How many times bigger is the volume of the sphere to the right?
r = 3
r = 9
Example:
How many times bigger is the radius of the sphere to the right?3 times bigger
9 times bigger
27 times bigger
Comparing Similar Figures
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SAsmall = 4π(3)2 = 36π units2
SAbig = 4π(9)2 = 324π units2
Vsmall = 4/3 π(3)3 = 36π units3
Vbig = 4/3 π(9)3 = 972π units3
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123 The scale factor of 2 similar pyramids is 4. If the surface area of the larger one is 64 units2, what is surface area of the smaller one?
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124 The scale factor of 2 similar right square pyramids is 3. If the area of the base of the larger one is 36 u2 and its height is 12, what is the volume of the smaller one?
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125 An architect builds a scale model of a home using a scale of 2 in to 5 ft. Given the view of the roof of the model, how much roofing material is needed for the house?
12 in
6 in8 in
5 in 4 in3 in
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PARCC Sample Questions
The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions.
Good Luck!
Return to Table of Contents
Slide 300 / 311
Question 6/11Daniel buys a block of clay for an art project. The block is shaped like a cube with edge lengths of 10 inches.
Daniel decides to cut the block of clay into two pieces. He places a wire across the diagonal of one face of the cube, as shown in the figure. Then he pulls the wire straight back to create two congruent chunks of clay.
PARCC Released Question - PBA - Calculator Section
Topic: Intro to 3-D Solids
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126 Part A - Question #1: Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting on the surface he exposed when he cut through the cube. Determine the newly exposed two-dimensional cross section.
A TriangleB ParallelogramC RectangleD RhombusE Square
Question 6/11
PARCC Released Question - PBA - Calculator Section
Topic: Intro to 3-D Solids
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127 Part A - Question #2:Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting on the surface he exposed when he cut through the cube. Find the area of this newly exposed two-dimensional cross section. Round your answer to the nearest whole square inch.
Question 6/11
PARCC Released Question - PBA - Calculator Section
Topic: Intro to 3-D Solids
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128 Part B:Daniel wants to reshape the other chunk of clay to make a set of clay spheres. He wants each sphere to have a diameter of 4 inches. Find the maximum number of spheres that Daniel can make from the chunk of clay. Show your work.
Question 6/11
PARCC Released Question - PBA - Calculator Section
Topic: Cavaleri's Principle
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Question 10/11The Farmer Supply is building a storage building for fertilizer that has a cylindrical base and a cone-shaped top. The county laws say that the storage building must have a maximum width of 8 feet and a maximum height of 14 feet.
Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone
PARCC Released Question - PBA - Calculator Section
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129 Dump Trucks deliver fertilizer in loads that are 4 feet tall, 6 feet wide & 12 feet long. Farmer Supply wants to be able to store 2 dump-truck loads of fertilizer.Determine the height of the cylinder, h1, and a height of the cone, h2, that Farmer Supply should use in the design. Show that your design will be able to store at least two dump-truck loads of fertilizer. When you finish, enter your value for h1 into your Responder.
Question 10/11 Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone
PARCC Released Question - PBA - Calculator Section - SMART Response Format
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130 A rectangle will be rotated 360º about a line which contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result of the rotation?
A a cube
B a rectangular prism
C a cylinder
D a sphere
Question 4/7
PARCC Released Question - EOY - Non-Calculator Section
Topic: Intro to 3-D Solids
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131 The table shows the approximate measurements of the Great Pyramid of Giza in Egypt and the Pyramid of Kukulcan in Mexico.
Approximately, what is the difference between the volume of the Great Pyramid of Giza and the volume of the Pyramid of Kukulcan?
A 1,945,000 cubic meters
B 2,562,000 cubic meters
C 5,835,000 cubic meters
D 7,686,000 cubic meters
PARCC Released Question - EOY - Calculator Section
Topic: Volume of a PyramidQuestion 8/25
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Question 11/25Two cylinders each with a height of 50 inches are shown.
Topic: Cavaleri's Principle
PARCC Released Question - EOY - Calculator Section
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132 Which statements about cylinders P and S are true? Select all that apply.A If x = y, the volume of cylinder P is greater than the volume
of cylinder S, because cylinder P is a right cylinder.B If x = y, the volume of cylinder P is equal to the volume of
cylinder S, because the cylindres are the same height.C If x = y, the volume of cylinder P is less than the volume of
cylinder S, because cylinder S is slanted.D If x < y, the area of a horizontal cross section of cylinder P is
greater than the area of a horizontal cross section of cylinder S.
E If x < y, the area of a horizontal cross section of cylinder P is equal to the area of a horizontal cross section of cylinder S.
F If x < y, the area of a horizontal cross section of cylinder P is less than the area of a hoizontal cross section of cylinder S.
Question 11/25 Topic: Cavaleri's Principle
PARCC Released Question - EOY - Calculator Section
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133 Part AThe outer surface of the pipe is coated with protective material. How many square feet is the outer surface of the pipe? Give your answer to the nearest integer.
A steel pipe in the shape of a right circular cylinder is used for drainage under a road. The length of the pipe is 12 feet and its diameter is 36 inches. The pipe is open at both ends.
Question 13/25 Topic: Surface Area of a Cylinder
PARCC Released Question - EOY - Calculator Section
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134 Part BA wire screen in the shape of a square is attached at one end of the pipe to allow water to flow through but to keep people from wandering into the pipe. The length of the diagonals of the screen are equal to the diameter of the pipe. The figure represents the placement of the screen at the end of the pipe.
A 72 B 102 C 125
D 324 E 648 F 1,018and the area of the screen is ________ square inches.
Question 13/25 Topic: Surface Area of a Cylinder
PARCC Released Question - EOY - Calculator Section
The perimeter of the screen is approximately ________ inches,
Select from each set of answers to correctly complete the sentence.