surface area volumecontent.njctl.org/courses/math/geometry/3d...jun 19, 2011 · progressive...

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Surface Area&

Volume

www.njctl.org

March 7, 2012

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Table of Contents1. Drawing 3-D Figures2. Surface v. Solid 3. Right v. Oblique

4. Nets

5. Views

6. Surface Area of a Prism

7. Surface Area of a Cylinder

8. Surface Area of a Pyramid

9. Surface Area of a Cone

10. Spheres

11. Surface Area of a Sphere

Click on the topic to go to that section

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Understanding 3-Dimensional Drawings

Return to Table of Contents

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X

Y

Y

X

2-dimensional drawings use only the x and y axes

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X

Y

Length

widthY

X

Length width

Y

X Length

width

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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

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Y

X

Z

height

height

Y

X

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Y

X

Z

height

height

YX

x

Y

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Y

X

Z

height

Y

X

X

Y

r

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To give a figure more of a 3-dimensional, 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

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The 3-Dimensional Figures discussed in this unit are:

PrismsPyramidsCylinders

ConesSpheres

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Prisms have 2 congruent polygon bases. The sides of a base are called base edges.The segments connecting corresponding vertices are lateral edges. A

B

C

XY

Z

In this diagram:There 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.

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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. A

B

C

XY

Z

In this diagram:There 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.

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1

A

B

C

D

E

F

G

H

I

A B CD

EF

M

N PQ

RS

Choose all of the base edges.

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2

A

B

C

D

E

F

G

H

I

A B CD

EF

M

N PQ

RS

Choose all of the lateral edges.

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3

A

B

C

D

E

F

G

H

I

A B CD

EF

M

N PQ

RS

Choose all of the edges.

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4 Chooses all of the bases.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

A B CD

EF

M

N PQ

RS

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5 Chooses all of the lateral faces.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

A B CD

EF

M

N PQ

RS

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6 Chooses all of the faces.

A AFSM

B FERS

C EDQR

D ABCDEF

E CDQP

F BCPN

G MNPQRS

H ABNM

A B CD

EF

M

N PQ

RS

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A pyramid has 1 base and the lateral edges go to a single point. This point is called the vertex.

A

M

N PQ

RS

This pyramid has: 6 lateral edges, 6 base edges, 12 edges (total)

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A pyramid is 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)

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7 Choose all of the base edges.

A

B

C

D

E

F

KL

MN

V

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8 Choose all of the base edges.

A

B

C

D

E

F

KL

MN

V

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9 How many edges does the pyramid have?

KL

MN

V

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10 Choose all of the lateral faces.

A KNV

B NMV

C KLMN

D VML

E KLV

KL

MN

V

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11 Choose all of the bases.

A KNV

B NMV

C KLMN

D VML

E KLV

KL

MN

V

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12 How many faces does the pyramid have?

KL

MN

V

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A cone, like a pyramid, has one base which is a circle.

. N

V

is thebase of the cone.

V is the vertex of the cone.

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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

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13 Which shape has 2 bases?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

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14 Which shape has a vertex?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

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15 Which shape has more base edges than lateral edges?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

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16 Which shape has no vertices?

A Prism

B Pyramid

C Cylinder

D Cone

E Sphere

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Surface v. Solid


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Surfaces and Solids are 3-dimensional figures.

A surface is the shell of a figure.

A solid is a filled figure.

In drawings, a solid is shaded so you cannot see

through it.

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17 Is the following object a solid or a surface? A Brick

A Solid

B Surface

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18 Is the following object a solid or a surface? An empty shoe box

A Solid

B Surface

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19 Is the following object a solid or a surface? A new can of soda

A Solid

B Surface

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20 Is the following object a solid or a surface? An ice cream cone

A Solid

B Surface

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A cross-section is the locus of points of the intersection of a plane and a space figure.

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Think about as if the plane were a knife and you were cutting the shape, what would the cut look like?

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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.)

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21 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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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 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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24 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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25 Choose all of the following that have cross-sections that consist of a 2-dimensional shape and its interior.

A a brick

B a balloon

C an empty soda can

D a stick of butter

E a wrapping paper tube

F a baseball

G a straw

H a book

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Right v. Oblique


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Oblique lines- two lines that intersect and are not

perpendicular.

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Right Triangular Prism Oblique Triangular Prism

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Right Cylinder Oblique Cylinder

How can a right prism or cylinder be distinguished from an oblique figure?

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Right Hexagonal Pyramid Oblique Hexagonal Pyramid

A right pyramid with a regular polygon for the base is called a regular pyramid. So this surface could also be called a regular hexagonal pyramid.

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Right Cone Oblique Cone

How can a right pyramid or cone be distinguished from an oblique figure?

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Right Triangular Prism Oblique Triangular Prism

What shape is each lateral face?

What shape is each lateral face?

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Naming a Figure

Rightor

Oblique

Shapeof

Base

Pyramidor

Prism

Solidor

Surface

Since cones and cylinders always circles as bases:

Rightor

Oblique

Solidor

Surface

Cylinderor

ConeA sphere is neither right or oblique:

Solidor

Surface

Spherical

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Right

Oblique

Triangular

Square

Rectangle

Pentagonal

Hexagonal

CylinderConePyramidPrism

Solid

Surface

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Right

Oblique

Triangular

Square

Rectangle

Pentagonal

Hexagonal


Solid

Surface

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Right

Oblique

Triangular

Square

Rectangle

Pentagonal

Hexagonal


Solid

Surface

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Right

Oblique

Triangular

Square

Rectangle

Pentagonal

Hexagonal


Solid

Surface

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Right

Oblique

Triangular

Square

Rectangle

Pentagonal

Hexagonal


Solid

Surface

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Nets


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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

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Shown is the net of a right rectangular prism.

Net

6

646 4

1212

64

4

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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.

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Net

This is a right square pyramid. Another name for it is pentahedron.Hedron is a suffice that means face.Why is this a pentahedron?

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radius

The net of a right cylinder is two circles and a rectangle that forms the lateral surface.

88

x

What is the length of x?

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Net

The lateral region is a circle with a sector missing. The bigger the slice missing will have what effect on the the cone?

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26 The net shown is for the given surface. Find x.

3

4

5

8

x

3

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3

4

5

8x

3

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3

4

5

8

x

3

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x8

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x

8

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Views


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A view is another type of 2 dimensional drawing of a 3 dimensional drawing.

The drawing depends on your position relative to the figure.

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Consider these three people viewing a pyramid:

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The orange person is standing in front of a face, so their view is a triangle.

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The green person is standing in front of a lateral edge, so from their view they can see 2 faces.

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The purple person is flying over and can see the four lateral faces.

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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 (front)

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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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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)

right square prism

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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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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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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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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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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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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?

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A

What would the view be like from each

position?

From A, how many columns of blocks are visible? How tall is each column?

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B


position?

From B, how many columns of blocks are visible? How tall is each column?

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C(Looking down from above)


position?

From C, how many piles of blocks are visible?

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FrontRight

AboveDraw the 3 views.

Right ViewTop View

Front View

Move for Answer

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FrontRight

AboveDraw the 3 views.

AboveFront RightMove for Answer

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Here are 3 views of a solid, draw a 3-dimensional representation.

Top FrontLeft

Move for Answer

L R

F

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Here are 3 views of a solid, draw a 3-dimensional representation.

Top

Left FrontF

L R

Move for Answer

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Surface Area of a Prism


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Rectangular Prisms

cube

LLL

www

h

hh

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Base

Baseheight

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

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The Surface Area of a figure is the total amount of Area that is needed to cover the entire figure

Area

Area

Area

Area

AreaArea

Top Area

Side Area

Front AreaBottom Area

Back Area

Side Area

The Surface Area of a figure is the sum of the areas of each side of the figure

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Finding the Surface Area of a Rectangular Prism

h

Lw

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. = 2Lw + 2Lh + 2 wh

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Example: Find the surface area of the prism

74

3

Area of Top

Area of Bottom

Area of Right Side

Area of Left Side

Area of Front

Area of Back

A=7(4) = 28u2

A=7(4) = 28u2

A=3(4) = 12 u2

A=3(4) = 12 u2

A=3(7) = 21 u2

A=3(7) = 21 u2

Total Surface Area = 28 + 28 +12 +12 + 21 + 21 = 122 u2

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40 What is the area of the top, in square units?

4

5

9

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41 What is the area of the right side, in square units?

4

5

9

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42 What is the area of the front, in square units?

4

5

9

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43 What is the total surface area, in square units?

4

5

9

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44 What is the area of the top, in square units?

8

8

8

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45 What is the area of the right side, in square units?

8

8

8

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46 What is the area of the front, in square units?

8

8

8

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47 What is the total surface area, in square units?

8

8

8

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48 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 (PH)

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

height

Lw

h

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 = 2L + 2w H = Height of the prism

S.A. = 2B + PH

S.A. = 2Lw + (2L +2w)H

S.A. = 2Lw + 2LH + 2wH

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Base

Base

height

Lw

h

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 = 2L + 2w H = Height of the prism

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 Area

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49 The surface area of the rectangular prism is :

A 24 sq ftB 144 sq ftC 288 sq ft

D 48 sq ftE 72 sq ft

12 ft6 ft

4 ft

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50 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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51 What is the total square units of the surface area?

7

96

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52 If 7 by 6 is base of the prism, what is the lateral area, in sq units?

7

96

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53 Find y, if the lateral area is 144 sq units. y by 6 is the base.

y

6 8

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54 What is the value of the missing variable if the surface area is 350 sq. ft.

A 7 ft

B 15 ft

C 17 ft

D 12 ft

X ft

5 ft

10 ft

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Triangular 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 Triangular Prism is a Triangle

The Height of the Prism is the length between the two Triangular Bases

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Area AreaArea

Area

Area

Area AreaArea

Area Area

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Finding the Surface Area of a Right Triangular 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 = H

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 heighta

b

c

H

P = a + b + c

ac

bc a

Lateral Surface

h

b

B = 1/2 bh

H

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Finding the Surface Area of a Right Triangular 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 = H

Lateral Area = PH = (a + b + c)H

The formula for a right triangular prism is the same as the formula for a right rectangular. This formula 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

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Example: Find the lateral area and surface area of the right triangular prism.

99

9

12

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55 Find the lateral area of the right prism.

8

76

9

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5

56

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4

20o 9

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58 The height of the triangular prism below is 11 ft, the base height is 3 ft, and the triangular base is an isosceles triangle. Find the surface area.

A 88 sq ft

B 100 sq ft

C 112 sq ft

D 125 sq ft3 ft

5 ft11 ft

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59 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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60 The right triangular prism has a surface area of 150 sq ft. Find the height of the prism.

A 5 ft

B 6 ft

C 7.81 ft

D 6.38 ft

65

y

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Example: Find the lateral area and surface area of the right prism.

Angles are right angles.

83

7

6

5

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Example: Find the lateral area and surface area of the right prism.

The base is a regular hexagon.

8

11

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4

4 32

10

9

All angles are right angles.

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62 Find the total surface area of the right prism.

4

4 32

10

9

All angles are right angles.

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The base is a regular pentagon.

6 ft

10 ft

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64 Find the total surface area of the right prism.

The base is a regular pentagon.

6 ft

10 ft

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Surface Area of a Cylinder


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Cylinders

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height

radius

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base

base

heightbase

base

height

A prism has 2 Bases

The Base of a Cylinder is a Circle

The Height of the prism is the length between the two Circular Bases

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Area

Area

AreaArea

Area

Area

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Finding the Surface Area of a Right CylinderSurface 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 = H

Lateral Area = CH = 2 π r H

The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the circular bases.

Base

Base

height

Base

Base

heightLateral Surface

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Example: Find the lateral area and surface area of the right cylinder.

8

r = 4

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34

d= 16

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Base Circumference is 16π ftHeight is 10 ft

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65 Find the lateral area of the right cylinder.

h = 12

r = 7

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66 Find the surface area of the right cylinder and round to two decimal places.

A 1200 sq in.

B 307.72 sq in.

C 835.24 sq in.

D 1670.48 sq in.

h = 12

r = 7

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r = 5

13

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h = 12

Base area is 36π units 2

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h = 12

Base area is 36π units 2

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70 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder.

A 7

B 8

C 5

D 6r = 8

h

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Surface Area of a Pyramid


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Pyramids

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Heightof theTriangle

Slant Height

The Pyramid has a square base and 4 triangular faces

The 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.

Surface Area = Sum of the Areas of all the sides

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Square Base (B)

Slant Height (l )

Pyramid's Height (h)

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Example: Find the value of x.

x

1312

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Example: Find the value of x.

Base Area of the right square pyramid is 64 u2.

x8

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Example: Find the length of the slant height.

This is a regular hexagonal pyramid.

r=6lateral edge= 12

r

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71 Find the value of the variable.

16

x6

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12

11x

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area of the base is 36 u2

x6

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74 Find the value of the slant height.

r

r= 8lateral edge = 15

Regular Hexagonal Pyramid

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75 Find the value of the slant height.

a

a= 9lateral edge = 12

Regular Hexagonal Pyramid

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Finding the Surface Area of a Regular Pyramid

Square Base (B)

Slant Height (l )

Pyramid's Height (h)

Surface Area = B + ½Pl Lateral Area = ½Pl

l = Slant HeightP = Perimeter of BaseB = Area of Base

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Example: Find the lateral area and the surface area of the pyramid.

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a

a=4lateral edge= 8

Regular Pentagonal Pyramid

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76 Find the lateral area of the right pyramid.

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77 Find the surface area of the right pyramid.

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Regular Octagonal Pyramid

a

a=5h=12

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Regular Octagonal Pyramid

a

a=5h=12

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30

12

8

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Surface Area of a Cone


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r

height

Slant Height l

Base

Lateral Surface

Slant Height l

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.

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Base

Lateral Surface

Slant Height l

Surface Area = Area of the Base + Lateral AreaLateral Area= ½PlS.A. = B + ½Pl

l = Slant HeightP = Perimeter of Circular BaseB = Area of Circular Base

Because the base is a circle.

L.A. = πrl S.A. = πr2 + πrl

Finding the Surface Area of a Right Cone

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Example: Find the lateral area and surface area of the right cone.

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Example: Find the lateral area and surface area of the right cone.

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83 Find the lateral area of the right cone, in square units.

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84 Find the surface area of the right cone, in square units.

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85 Find the lateral area of the right cone, in square units.

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86 Find the surface area of the right cone, in square units.

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87 Find the length of the radius of the right cone if the lateral area is 50π units2?

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88 Find the height of the right cone if the lateral area is 50π units2?

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89 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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90 The Department of Transportation keeps piles of road salt for snowy days. The conical shaped piles are 20 feet high and 30 feet across at the base. During the summer the piles are covered with tarps to prevent erosion. How many square feet of tarp is needed so that no part of the pile is exposed?

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Spheres

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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

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Sphere

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

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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.

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Y

X

Z

Great Circles

Each of these planes intersects the sphere, and the plane contains the center of the sphere

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HemisphereThe 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 SectionsA 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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The farther the cross section of the sphere is taken from its center the smaller the circle.

.

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Example: Find the radius of the cross section of the sphere with radius 8 if the cross section is 2 from the center.

82

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Example: Find the radius of the cross section of the sphere with radius 8 if the cross section is 2 from the center.

82

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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?

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91 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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92 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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93 The area of the great circle of a sphere is 12π units2 and a cross section has area 8π units2. How far is the cross section from the center?

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Surface Area of a Sphere


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Sphere

radius

Center

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r S.A. = 4πr2

Finding the Surface Area of the Sphere

Why is there no formula for lateral area?

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Example: Find the surface area of a sphere with radius 6 ft.

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Example: Find the surface area of a sphere that a great circle with area 24π?

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Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is surface area of the sphere?

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94 Find the surface area of a sphere with radius 10.

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95 What is the surface area of a sphere if a cross section 7 units from the center has an area of 50 units2?

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96 The surface area of a sphere is 24 units. What is the area of a great circle of a congruent sphere?

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97 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 area that Shelly has to cover?

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surface area volumecontent.njctl.org/courses/math/geometry/3d...jun 19, 2011 · progressive...

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