the geometry of three dimensions
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Geometry Chap 11. The Geometry of Three Dimensions. Eleanor Roosevelt High School Chin-Sung Lin. ERHS Math Geometry. The Geometry of Three Dimensions. The geometry of three dimensions is called solid geometry. Mr. Chin-Sung Lin. ERHS Math Geometry. Points, Lines, and Planes. - PowerPoint PPT PresentationTRANSCRIPT
The Geometry of Three Dimensions
Eleanor Roosevelt High School Chin-Sung Lin
Geometry Chap 11
The Geometry of Three Dimensions
The geometry of three dimensions is called solid geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Points, Lines, and Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates of the Solid Geometry
There is one and only one plane containing three non-collinear points
Mr. Chin-Sung Lin
ERHS Math Geometry
A
B
C
Postulates of the Solid Geometry
A plane containing any two points contains all of the points on the line determined by those two points
Mr. Chin-Sung Lin
ERHS Math Geometry
m A
B
Theorems of the Points, Lines & Planes
There is exactly one plane containing a line and a point not on the line
Mr. Chin-Sung Lin
ERHS Math Geometry
mA
B
P
Theorems of the Points, Lines & Planes
If two lines intersect, then there is exactly one plane containing them
Two intersecting lines determine a plane
Mr. Chin-Sung Lin
ERHS Math Geometry
m
AB P
n
Parallel Lines in Space
Lines in the same plane that have no points in common
Two lines are parallel if and only if they are coplanar and have no points in common
Mr. Chin-Sung Lin
ERHS Math Geometry
m
n
Skew Lines in Space
Skew lines are lines in space that are neither parallel nor intersecting
Mr. Chin-Sung Lin
ERHS Math Geometry
m
n
Example
Both intersecting lines and parallel lines lie in a plane
Skew lines do not lie in a plane
Identify the parallel lines, intercepting lines, and skew lines in the cube
Mr. Chin-Sung Lin
ERHS Math Geometry
A B
D C
E F
H G
Perpendicular Lines and Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
Postulates of the Solid Geometry
If two planes intersect, then they intersect in exactly one line
Mr. Chin-Sung Lin
ERHS Math Geometry
A
B
Dihedral Angle
A dihedral angle is the union of two half-planes with a common edge
Mr. Chin-Sung Lin
ERHS Math Geometry
The Measure of a Dihedral AngleThe measure of the plane angle formed by two rays
each in a different half-plane of the angle and each perpendicular to the common edge at the same point of the edge
AC AB and AD AB The measure of the dihedral angle:
mCAD
Mr. Chin-Sung Lin
ERHS Math Geometry
C
A
B
D
Perpendicular PlanesPerpendicular planes are two planes that intersect to
form a right dihedral angle
AC AB, AD AB, and AC AD (mCAD = 90)thenm n
Mr. Chin-Sung Lin
ERHS Math Geometry
C
A
B
D
m
n
Theorems of Perpendicular Lines & Planes
If a line not in a plane intersects the plane, then it intersects in exactly one point
Mr. Chin-Sung Lin
ERHS Math Geometry
k
AB Pn
A Line is Perpendicular to a Plane
A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the intersection of the line and the plane
A plane is perpendicular to a line if the line is perpendicular to the plane
k m, and k n,
then k s
Mr. Chin-Sung Lin
ERHS Math Geometry
n
p
ks
m
Postulates of the Solid Geometry
At a given point on a line, there are infinitely many lines perpendicular to the given line
Mr. Chin-Sung Lin
ERHS Math Geometry
n
A
k
m
p qr
Theorems of Perpendicular Lines & Planes
If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by these lines
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
Connect AB
Connect PT and intersects AB at QMake PR = PS
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
Connect RA, SA
SASΔRAP = ΔSAP
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
CPCTC
AR = AS
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
Connect RB, SB
SASΔRBP = ΔSBP
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
CPCTC
BR = BS
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
SSS
ΔRAB = ΔSAB
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
CPCTC
RAB = SAB
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
Connect RQ, SQ
SASΔRAQ = ΔSAQ
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k m
CPCTC
QR = QS
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k mSSSΔRPQ = ΔSPQ
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k AP and k BP
Prove: k mCPCTCmRPQ = mSPQmRPQ + mSPQ = 180mRPQ = mSPQ = 90
Mr. Chin-Sung Lin
ERHS Math Geometry
A
P
km
B
R
S
QT
Theorems of Perpendicular Lines & Planes
If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane
Given: Plane p plane q
Prove: A line in p is perpendicular to q
and a line in q is perpendicular to p
Mr. Chin-Sung Lin
ERHS Math Geometry
A BD
p
q
C
Theorems of Perpendicular Lines & Planes
If a plane contains a line perpendicular to another plane, then the planes are perpendicular
Given: AC in plane p and AC q
Prove: p q
Mr. Chin-Sung Lin
ERHS Math Geometry
A BD
p
q
C
Theorems of Perpendicular Lines & Planes
Two planes are perpendicular if and only if one plane contains a line perpendicular to the other
Mr. Chin-Sung Lin
ERHS Math Geometry
A BD
p
q
C
Theorems of Perpendicular Lines & Planes
Through a given point on a plane, there is only one line perpendicular to the given plane
Given: Plane p and AB p at A
Prove: AB is the only line perpendicular to p at A
Mr. Chin-Sung Lin
ERHS Math Geometry
p A
B
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
p A
BC
D
Through a given point on a plane, there is only one line perpendicular to the given plane
Given: Plane p and AB p at A
Prove: AB is the only line perpendicular to p at A
q
Through a given point on a line, there can be only one plane perpendicular to the given line
Given: Any point P on AB
Prove: There is only one plane
perpendicular to AB
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
P
A
B
Through a given point on a line, there can be only one plane perpendicular to the given line
Given: Any point P on AB
Prove: There is only one plane
perpendicular to AB
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
A
B
RP nQ m
If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the plane
Given: AB p at A and AB AC
Prove: AC is in plane p
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
p A
B
CD
q
If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
Given: Plane p with AB p at A, and
C any point not on p
Prove: Plane q determined by A, B, and C
is perpendicular to p
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
p A
B C
q
If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
Given: Plane p with AB p at A, and
C any point not on p
Prove: Plane q determined by A, B, and C
is perpendicular to p
Theorems of Perpendicular Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
p A
B C
D
q
E
Parallel Lines and Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel Planes
Parallel planes are planes that have no points in common
Mr. Chin-Sung Lin
ERHS Math Geometry
m
n
A Line is Parallel to a Plane
A line is parallel to a plane if it has no points in common with the plane
Mr. Chin-Sung Lin
ERHS Math Geometry
k
m
Theorems of Parallel Lines & Planes
If a plane intersects two parallel planes, then the intersection is two parallel lines
Mr. Chin-Sung Lin
ERHS Math Geometry
n
m
p
Theorems of Parallel Lines & Planes
If a plane intersects two parallel planes, then the intersection is two parallel lines
Given: Plane p intersects plane m at AB and plane n at CD, m//n
Prove: AB//CD
Mr. Chin-Sung Lin
ERHS Math Geometry
n
mA B
C D
p
Theorems of Parallel Lines & Planes
Two lines perpendicular to the same plane are parallel
Given: Plane p, LA⊥p at A, and MB⊥p at B
Prove: LA//MB
Mr. Chin-Sung Lin
ERHS Math Geometry
p
q
L M
AB
Theorems of Parallel Lines & Planes
Two lines perpendicular to the same plane are parallel
Given: Plane p, LA⊥p at A, and MB⊥p at B
Prove: LA//MB
Mr. Chin-Sung Lin
ERHS Math Geometry
p
q
L M
AB
CD
N
Theorems of Parallel Lines & Planes
Two lines perpendicular to the same plane are coplanar
Given: Plane p, LA⊥p at A, and MB⊥p at B
Prove: LA and MB are coplanar
Mr. Chin-Sung Lin
ERHS Math Geometry
p
q
L M
AB
Theorems of Parallel Lines & Planes
If two planes are perpendicular to the same line, then they are parallel
Given: Plane p⊥AB at A and q⊥AB at B
Prove: p//q
Mr. Chin-Sung Lin
ERHS Math Geometry
qB
p
A
Theorems of Parallel Lines & Planes
If two planes are perpendicular to the same line, then they are parallel
Given: Plane p⊥AB at A and q⊥AB at B
Prove: p//q
Mr. Chin-Sung Lin
ERHS Math Geometry
q
p
A
BR
s
Theorems of Parallel Lines & Planes
If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other
Given: Plane p parallel to plane q, and AB⊥p and intersectingplane q at B
Prove: q⊥AB
Mr. Chin-Sung Lin
ERHS Math Geometry
qB
p
A
Theorems of Parallel Lines & Planes
If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other
Given: Plane p parallel to plane q, and AB⊥p and intersectingplane q at B
Prove: q⊥AB
Mr. Chin-Sung Lin
ERHS Math Geometry
B
A
E
C
q
p
If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other
Given: Plane p parallel to plane q, and AB⊥p and intersectingplane q at B
Prove: q⊥AB
Theorems of Parallel Lines & Planes
Mr. Chin-Sung Lin
ERHS Math Geometry
B
A
E
C
q
p
F
D
Theorems of Parallel Lines & Planes
Two planes are perpendicular to the same line if and only if the planes are parallel
Mr. Chin-Sung Lin
ERHS Math Geometry
qB
p
A
Distance between Two Planes
The distance between two planes is the length of the line segment perpendicular to both planes with an endpoint on each plane
Mr. Chin-Sung Lin
ERHS Math Geometry
B
A
q
p
Theorems of Parallel Lines & Planes
Parallel planes are everywhere equidistant
Given: Parallel planes p and q, with AC and BD each perpendicular to p and q with an endpoint on each plane
Prove: AC = BD
Mr. Chin-Sung Lin
ERHS Math Geometry
C
A B
q
pD
Surface Area of a Prism
Mr. Chin-Sung Lin
ERHS Math Geometry
Polyhedron
A polyhedron is a three-dimensional figure formed by the union of the surfaces enclosed by plane figures
A polyhedron is a figure that is the union of polygons
Mr. Chin-Sung Lin
ERHS Math Geometry
Polyhedron: Faces, Edges & Vertices
Faces: the portions of the planes enclosed by a plane figure
Edges: The intersections of the faces
Vertices: the intersections of the edges
Mr. Chin-Sung Lin
ERHS Math Geometry
Vertex
Edge
Face
Prism
A prism is a polyhedron in which two of the faces, called the bases of the prism, are congruent polygons in parallel planes
Mr. Chin-Sung Lin
ERHS Math Geometry
Prism: Lateral Sides, Lateral Edges, Altitude & Height
Lateral sides: the surfaces between corresponding sides of the bases
Lateral edges: the common edges of the lateral sides
Altitude: a line segment perpendicular to each of the bases with an endpoint on each base
Height: the length of an altitude
Mr. Chin-Sung Lin
ERHS Math Geometry
Lateral Side
Lateral Edge
Altitude/Height
Base
Prism: Lateral Edges
The lateral edges of a prism are congruent and parallel
Mr. Chin-Sung Lin
ERHS Math Geometry
Lateral Edges
Right Prism
A right prism is a prism in which the lateral sides are all perpendicular to the bases
All of the lateral sides of a right prism are rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Lateral Sides
Parallelepiped
A parallelepiped is a prism that has parallelograms as bases
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangular Parallelepiped
A rectangular parallelepiped is a parallelepiped that has rectangular bases and lateral edges perpendicular to the bases
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangular Solid
A rectangular parallelepiped is also called a rectangular solid, and it is the union of six rectangles. Any two parallel rectangles of a rectangular solid can be the bases
Mr. Chin-Sung Lin
ERHS Math Geometry
Area of a Prism
The lateral area of the prism is the sum of the areas of the lateral faces
The total surface area is the sum of the lateral area and the areas of the bases
Mr. Chin-Sung Lin
ERHS Math Geometry
Area of a Prism Example
Calculate the lateral area of the prism
Calculate the total surface area of the prism
Mr. Chin-Sung Lin
ERHS Math Geometry
4
7
5
Area of a Prism Example
Area of the bases: 7 x 5 x 2 = 70
Lateral area: 2 x (4 x 5 + 4 x 7) = 96
Total surface area: 70 + 96 = 166
Mr. Chin-Sung Lin
ERHS Math Geometry
4
75
Area of a Prism Example
The bases of a right prism are equilateral triangles
Calculate the lateral area of the prism
Calculate the total surface area of the prism
Mr. Chin-Sung Lin
ERHS Math Geometry
5
4
Area of a Prism Example
Area of the bases: ½ x (4 x 2√3) x 2= 8√3
Lateral area: 3 x (4 x 5) = 60
Total surface area: 60 + 8√3 ≈ 73.86
Mr. Chin-Sung Lin
ERHS Math Geometry
5
4
2
2√34
Volume of a Prism
Mr. Chin-Sung Lin
ERHS Math Geometry
Volume of a Prism
The volume (V) of a prism is equal to the area of the base (B) times the height (h)
V = B x h
Mr. Chin-Sung Lin
ERHS Math Geometry
Base (B)
Height (h)
Volume of a Prism Example
A right prism is shown in the diagram
Calculate the Volume of the prism
Mr. Chin-Sung Lin
ERHS Math Geometry
5
4
2
Volume of a Prism Example
A right prism is shown in the diagram
Calculate the Volume of the prism
B = ½ x 4 x 2 = 4
h = 5
V = Bh = 4 x 5 = 20
Mr. Chin-Sung Lin
ERHS Math Geometry
5
4
2
Volume of a Prism Example
A right prism is shown in the diagram
Calculate the Volume of the prism
Mr. Chin-Sung Lin
ERHS Math Geometry
3
5
4
Volume of a Prism Example
A right prism is shown in the diagram
Calculate the Volume of the prism
B = 5 x 4 = 20
h = 3
V = Bh = 20 x 3 = 60
Mr. Chin-Sung Lin
ERHS Math Geometry
3
5
4
Pyramids
Mr. Chin-Sung Lin
ERHS Math Geometry
Pyramids
A pyramid is a solid figure with a base that is a polygon and lateral faces that are triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Pyramids: Vertex & AltitudeVertex: All lateral edges meet in a point
Altitude: the perpendicular line segment from the vertex to thebase
Mr. Chin-Sung Lin
ERHS Math Geometry
Vertex
Altitude
Vertex
Altitude
Regular PyramidsA pyramid whose base is a regular polygon
and whose altitude is perpendicular to the base at its center
The lateral edges of a regular polygon are congruent
The lateral faces of a regular pyramid are isosceles triangles
The length of the altitude of a triangular lateral face is the slant height of the pyramid
Mr. Chin-Sung Lin
ERHS Math Geometry
Slant HeightAltitud
e
Surface Area of PyramidsThe lateral area of a pyramid is the sum of
the areas of the faces (isosceles triangles)
The total surface area is the lateral area plus the area of the base
Mr. Chin-Sung Lin
ERHS Math Geometry
Slant Height
Volume of PyramidsThe volume (V) of a pyramid is equal to one
third of the area of the base (B) times the height (h)
V = (1/3) x B x h
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Area
Height
Volume of Pyramids ExampleA regular pyramid has a square base. The
length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters
a. What is the total surface area of the pyramid?
b. What is the volume of the pyramid?
Mr. Chin-Sung Lin
ERHS Math Geometry
13
10
Volume of Pyramids ExampleA regular pyramid has a square base. The
length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters
a. What is the total surface area of the pyramid?
b. What is the volume of the pyramid?
Mr. Chin-Sung Lin
ERHS Math Geometry
13
10
Volume of Pyramids ExampleA regular pyramid has a square base. The
length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters
a. What is the total surface area of the pyramid?
b. What is the volume of the pyramid?
Mr. Chin-Sung Lin
ERHS Math Geometry
13
10
5
12
Volume of Pyramids Examplea. Total surface area:
Lateral Area: ½ x 10 x 13 x 4 = 260
Base Area: 10 x 10 = 100
Total Area = 260 + 100 = 360 cm2
b. Volume:
B = 100
h = 12
V = (1/3) x 100 x 12 = 400 cm3
Mr. Chin-Sung Lin
ERHS Math Geometry
13
10
5
12
Properties of Regular PyramidsThe base of a regular pyramid is a regular
polygon and the altitude is perpendicular to the base at its center
The center of a regular polygon is defined as the point that is equidistant to its vertices
The lateral faces of a regular pyramid are isosceles triangles
The lateral faces of a regular pyramid are congruent
Mr. Chin-Sung Lin
ERHS Math Geometry
Cylinders
Mr. Chin-Sung Lin
ERHS Math Geometry
Cylinders
The solid figure formed by the congruent parallel curves and the surface that joins them is called a cylinder
Mr. Chin-Sung Lin
ERHS Math Geometry
CylindersBases: the closed curves
Lateral surface: the surface that joins the bases
Altitude: a line segment perpendicular to the bases with endpoints on the bases
Height: the length of an altitude
Mr. Chin-Sung Lin
ERHS Math Geometry
BasesLateral Surface
Altitude
Circular CylindersA cylinder whose bases are congruent circles
Mr. Chin-Sung Lin
ERHS Math Geometry
Right Circular CylindersIf the line segment joining the centers of the circular
bases is perpendicular to the bases, the cylinder is a right circular cylinder
Mr. Chin-Sung Lin
ERHS Math Geometry
Surface Area of Right Circular Cylinders
Base Area: 2πr2
Lateral Area: 2πrh
Total Surface Area: 2πrh + 2πr2
Mr. Chin-Sung Lin
ERHS Math Geometry
r
h
Volume of Circular Cylinders
Volume: B x h = πr2h
Mr. Chin-Sung Lin
ERHS Math Geometry
Right Circular Cylinders ExampleA right cylinder as shown in the diagram.
Calculate the total Surface Area
Calculate the volume
Mr. Chin-Sung Lin
ERHS Math Geometry
6
14
Right Circular Cylinders ExampleBase Area:
2πr2 = 2π62 ≈ 226.19
Lateral Area:
2πrh = 2π (6)(14) ≈ 527.79
Total Surface Area:
226.19 + 527.79 = 754.58
Volume:
B x h = πr2h = π(62)(14) = 1583.36
Mr. Chin-Sung Lin
ERHS Math Geometry
6
14
Cones
Mr. Chin-Sung Lin
ERHS Math Geometry
Right Circular Conical Surface
Line OQ is perpendicular to plane p at O, and a point P is on plane p
Keeping point Q fixed, move P through a circle on p with center at O. The surface generated by PQ is a right circular conical surface
* A conical surface extends infinitely
Mr. Chin-Sung Lin
ERHS Math Geometry
A
CP
O
Q
p
Right Circular ConeThe part of the conical surface
generated by PQ from plane p to Q is called a right circular cone
Q: vertex of the cone
Circle O: base of the cone
OQ: altitude of the cone
OQ: height of the cone, and
PQ: slant height of the cone
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
CP
O
Q
p
Surface Area of a ConeBase Area: B = πr2
Lateral Area: L = ½ Chs= ½ (2πr)hs = πrhs
Total Surface Area: πrhs + πr2
* hs: slant height
* hc: height
* r: radius* B: base area* C: circumference
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
C
hs
C
r
p
hc
B
Volume of a ConeBase Area: B = πr2
Volume: V = ⅓ Bhc= ⅓ πr2hc
* hs: slant height
* hc: height
* r: radius* B: base area* C: circumference
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
C
hs
B r
p
hc
C
Surface Area of a Cone ExampleCalculate the base area, lateral area, and
total area
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
C
26
10
p
24
Surface Area of a Cone ExampleCalculate the base area, lateral area, and
total area
Base Area: B = π(10)2 = 100π
Lateral Area: L = π(10)(26) = 260π
Total Surface Area: 100π + 260π
= 360π
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
C
26
10
p
24
Volume of a Cone ExampleA cone and a cylinder have equal volumes
and equal heights. If the radius of the base of the cone is 3 centimeters, what is the radius of the base of the cylinder?
Volume of Cylinder: V = h = πr2h
Volume of Cone: V = ⅓ π32h = 3πh
πr2h = 3πh, r2 = 3, r = √3 cm
Mr. Chin-Sung Lin
ERHS Math GeometryA
A
C3 cm
p
h
r
h
Spheres
Mr. Chin-Sung Lin
ERHS Math Geometry
Spheres
A sphere is the set of all points equidistant from a fixed point called the center
The radius of a sphere is the length of the line segment from the center of the sphere to any point on the sphere
Mr. Chin-Sung Lin
ERHS Math Geometry
rO
Sphere and PlaneIf the distance of a plane from the center of a sphere is
dand the radius of the sphere is r
Mr. Chin-Sung Lin
ERHS Math Geometry
P
O
p
dr
P
O
p
dr
P
O
p
dr
r < d no points in common
r = d one points in common
r > d infinite points
in common (circle)
CirclesA circle is the set of all points in a plane equidistant
from a fixed point in the plane called the center
Mr. Chin-Sung Lin
ERHS Math Geometry
Op
r
Theorem about CirclesThe intersection of a sphere and a plane through the
center of the sphere is a circle whose radius is equal to the radius of the sphere
Mr. Chin-Sung Lin
ERHS Math Geometry
O
p
r
r
Great Circle of a Sphere A great circle of a sphere is the intersection of a
sphere and a plane through the center of the sphere
Mr. Chin-Sung Lin
ERHS Math Geometry
O
p
r
r
Theorem about CirclesIf the intersection of a sphere and a plane does not contain
the center of the sphere, then the intersection is a circle
Given: A sphere with center at O plane p intersecting the sphere at A and B
Prove: The intersection is a circle
Mr. Chin-Sung Lin
ERHS Math Geometry
O
pCA
B
Theorem about CirclesIf the intersection of a sphere and a plane does not contain
the center of the sphere, then the intersection is a circle
Given: A sphere with center at O plane p intersecting the sphere at A and B
Prove: The intersection is a circle
Mr. Chin-Sung Lin
ERHS Math Geometry
O
p
rCA
B
Theorem about Circles
Mr. Chin-Sung Lin
Statements Reasons
1. Draw a line OC, point C on plane p 1. Given, create two triangles OCAC, OCBC 2. OCA and OCB are right angles 2. Definition of perpendicular3. OA OB 3. Radius of a sphere4. OC OC 4. Reflexive postulate5. OAC OBC 5. HL postulate6. CA CB 6. CPCTC7. The intersection is a circle 7. Definition of circles
ERHS Math Geometry
O
p
rCA B
Theorem about CirclesThe intersection of a plane and a sphere is a circle
A great circle is the largest circle that can be drawn on a sphere
Mr. Chin-Sung Lin
ERHS Math Geometry
O
p
p’
Theorem about CirclesIf two planes are equidistant from the center of a sphere and
intersect the sphere, then the intersections are congruent circles
Mr. Chin-Sung Lin
ERHS Math Geometry
O
q
p
A
B
C
D
Surface Area of a Sphere
Surface Area: S = 4πr2
r: radius
Mr. Chin-Sung Lin
ERHS Math GeometryA
rO
Volume of a Sphere
Volume: V = 4/3 πr3
r: radius
Mr. Chin-Sung Lin
ERHS Math GeometryA
rO
Sphere Example
Find the surface area and the volume of a sphere whose radius is 6 cm
Mr. Chin-Sung Lin
ERHS Math GeometryA
rO
Sphere Example
Find the surface area and the volume of a sphere whose radius is 6 cm
Surface Area: S = 4π62 = 144π cm2
Volume: V = 4/3 π63 = 288π cm3
Mr. Chin-Sung Lin
ERHS Math GeometryA
rO
Q & A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin
ERHS Math Geometry