The Geometry of Three Dimensions

1
The Geometry of Three Dimensions
Geometry Chap 11

• Eleanor Roosevelt High School
• Chin-Sung Lin

2
The Geometry of Three Dimensions
ERHS Math Geometry
The geometry of three dimensions is called
solid geometry
Mr. Chin-Sung Lin
3
Points, Lines, and Planes
ERHS Math Geometry
Mr. Chin-Sung Lin
4
Postulates of the Solid Geometry
ERHS Math Geometry
There is one and only one plane containing three
non-collinear points
B
A
C
Mr. Chin-Sung Lin
5
Postulates of the Solid Geometry
ERHS Math Geometry
A plane containing any two points contains all of
the points on the line determined by those two
points
B
A
m
Mr. Chin-Sung Lin
6
Theorems of the Points, Lines Planes
ERHS Math Geometry
There is exactly one plane containing a line and
a point not on the line
B
m
A
P
Mr. Chin-Sung Lin
7
Theorems of the Points, Lines Planes
ERHS Math Geometry
If two lines intersect, then there is exactly one
plane containing them Two intersecting lines
determine a plane
m
P
A
B
n
Mr. Chin-Sung Lin
8
Parallel Lines in Space
ERHS Math Geometry
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
m
n
Mr. Chin-Sung Lin
9
Skew Lines in Space
ERHS Math Geometry
Skew lines are lines in space that are neither
parallel nor intersecting
n
m
Mr. Chin-Sung Lin
10
Example
ERHS Math Geometry
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
11
Perpendicular Lines and Planes
ERHS Math Geometry
Mr. Chin-Sung Lin
12
Postulates of the Solid Geometry
ERHS Math Geometry
If two planes intersect, then they intersect in
exactly one line
B
A
Mr. Chin-Sung Lin
13
Dihedral Angle
ERHS Math Geometry
A dihedral angle is the union of two half-planes
with a common edge
Mr. Chin-Sung Lin
14
The Measure of a Dihedral Angle
ERHS Math Geometry
The 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 m?CAD
Mr. Chin-Sung Lin
15
Perpendicular Planes
ERHS Math Geometry
Perpendicular planes are two planes that
intersect to form a right dihedral angle AC ?
AB, AD ? AB, and AC ? AD (m?CAD 90) then m ?
n
m
n
Mr. Chin-Sung Lin
16
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
If a line not in a plane intersects the plane,
then it intersects in exactly one point
k
P
A
B
n
Mr. Chin-Sung Lin
17
A Line is Perpendicular to a Plane
ERHS Math Geometry
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
k
n
s
m
p
Mr. Chin-Sung Lin
18
Postulates of the Solid Geometry
ERHS Math Geometry
At a given point on a line, there are infinitely
many lines perpendicular to the given line
q
p
k
r
n
m
A
Mr. Chin-Sung Lin
19
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
A
m
P
B
Mr. Chin-Sung Lin
20
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
A
m
P
B
Mr. Chin-Sung Lin
21
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 Q Make PR PS
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
22
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 SAS ?RAP ?SAP
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
23
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
24
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 SAS ?RBP ?SBP
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
25
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
26
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 ?RAB ?SAB
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
27
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
28
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 SAS ?RAQ ?SAQ
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
29
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
30
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 ?RPQ ?SPQ
k
R
m
A
Q
T
P
B
S
Mr. Chin-Sung Lin
31
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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 m?RPQ
m?SPQ m?RPQ m?SPQ 180 m?RPQ m?SPQ 90
k
R
m
A
Q
T
B
P
S
Mr. Chin-Sung Lin
32
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
C
p
A
B
q
D
Mr. Chin-Sung Lin
33
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
C
p
A
B
q
D
Mr. Chin-Sung Lin
34
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
Two planes are perpendicular if and only if one
plane contains a line perpendicular to the other
C
p
A
B
q
D
Mr. Chin-Sung Lin
35
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
B
A
p
Mr. Chin-Sung Lin
36
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
B
C
A
D
p
Mr. Chin-Sung Lin
37
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
A
P
B
Mr. Chin-Sung Lin
38
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
A
Q
m
n
P
R
B
Mr. Chin-Sung Lin
39
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
q
B
C
A
D
p
Mr. Chin-Sung Lin
40
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
q
B
C
A
p
Mr. Chin-Sung Lin
41
Theorems of Perpendicular Lines Planes
ERHS Math Geometry
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
q
B
C
A
D
p
E
Mr. Chin-Sung Lin
42
Parallel Lines and Planes
ERHS Math Geometry
Mr. Chin-Sung Lin
43
Parallel Planes
ERHS Math Geometry
Parallel planes are planes that have no points in
common
m
n
Mr. Chin-Sung Lin
44
A Line is Parallel to a Plane
ERHS Math Geometry
A line is parallel to a plane if it has no points
in common with the plane
k
m
Mr. Chin-Sung Lin
45
Theorems of Parallel Lines Planes
ERHS Math Geometry
If a plane intersects two parallel planes, then
the intersection is two parallel lines
p
Mr. Chin-Sung Lin
46
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
p
B
A
D
C
Mr. Chin-Sung Lin
47
Theorems of Parallel Lines Planes
ERHS Math Geometry
Two lines perpendicular to the same plane are
parallel Given Plane p, LA?p at A, and MB?p at B
Prove LA//MB
q
M
L
B
A
p
Mr. Chin-Sung Lin
48
Theorems of Parallel Lines Planes
ERHS Math Geometry
Two lines perpendicular to the same plane are
parallel Given Plane p, LA?p at A, and MB?p at B
Prove LA//MB
q
N
M
L
B
A
D
C
p
Mr. Chin-Sung Lin
49
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
q
M
L
B
A
p
Mr. Chin-Sung Lin
50
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
A
p
B
q
Mr. Chin-Sung Lin
51
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
s
A
p
B
R
q
Mr. Chin-Sung Lin
52
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
intersecting plane q at B Prove q?AB
A
p
B
q
Mr. Chin-Sung Lin
53
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
intersecting plane q at B Prove q?AB
C
A
p
E
B
q
Mr. Chin-Sung Lin
54
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
intersecting plane q at B Prove q?AB
C
A
D
p
E
B
q
F
Mr. Chin-Sung Lin
55
Theorems of Parallel Lines Planes
ERHS Math Geometry
Two planes are perpendicular to the same line if
and only if the planes are parallel
A
p
B
q
Mr. Chin-Sung Lin
56
Distance between Two Planes
ERHS Math Geometry
The distance between two planes is the length of
the line segment perpendicular to both planes
with an endpoint on each plane
A
p
B
q
Mr. Chin-Sung Lin
57
Theorems of Parallel Lines Planes
ERHS Math Geometry
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
B
A
p
D
C
q
Mr. Chin-Sung Lin
58
Surface Area of a Prism
ERHS Math Geometry
Mr. Chin-Sung Lin
59
Polyhedron
ERHS Math Geometry
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
60
Polyhedron Faces, Edges Vertices
ERHS Math Geometry
Faces the portions of the planes enclosed by a
plane figure Edges The intersections of the
faces Vertices the intersections of the edges
Vertex
Edge
Face
Mr. Chin-Sung Lin
61
Prism
ERHS Math Geometry
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
62
Prism Lateral Sides, Lateral Edges, Altitude
Height
ERHS Math Geometry
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
Base
Lateral Side
Lateral Edge
Altitude/Height
Mr. Chin-Sung Lin
63
Prism Lateral Edges
ERHS Math Geometry
The lateral edges of a prism are congruent and
parallel
Lateral Edges
Mr. Chin-Sung Lin
64
Right Prism
ERHS Math Geometry
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
Lateral Sides
Mr. Chin-Sung Lin
65
Parallelepiped
ERHS Math Geometry
A parallelepiped is a prism that has
parallelograms as bases
Mr. Chin-Sung Lin
66
Rectangular Parallelepiped
ERHS Math Geometry
A rectangular parallelepiped is a parallelepiped
that has rectangular bases and lateral edges
perpendicular to the bases
Mr. Chin-Sung Lin
67
Rectangular Solid
ERHS Math Geometry
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
68
Area of a Prism
ERHS Math Geometry
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
69
Area of a Prism Example
ERHS Math Geometry
Calculate the lateral area of the prism Calculate
the total surface area of the prism
4
5
7
Mr. Chin-Sung Lin
70
Area of a Prism Example
ERHS Math Geometry
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
71
Area of a Prism Example
ERHS Math Geometry
The bases of a right prism are equilateral
triangles Calculate the lateral area of the
prism Calculate the total surface area of the
prism
5
4
Mr. Chin-Sung Lin
72
Area of a Prism Example
ERHS Math Geometry
Area of the bases ½ x (4 x 2v3) x 2 8v3 Lateral
area 3 x (4 x 5) 60 Total surface area 60
8v3 73.86
4
2
5
4
Mr. Chin-Sung Lin
73
Volume of a Prism
ERHS Math Geometry
Mr. Chin-Sung Lin
74
Volume of a Prism
ERHS Math Geometry
The volume (V) of a prism is equal to the area of
the base (B) times the height (h) V B x h
Base (B)
Height (h)
Mr. Chin-Sung Lin
75
Volume of a Prism Example
ERHS Math Geometry
A right prism is shown in the diagram Calculate
the Volume of the prism
2
5
4
Mr. Chin-Sung Lin
76
Volume of a Prism Example
ERHS Math Geometry
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
2
5
4
Mr. Chin-Sung Lin
77
Volume of a Prism Example
ERHS Math Geometry
A right prism is shown in the diagram Calculate
the Volume of the prism
4
3
5
Mr. Chin-Sung Lin
78
Volume of a Prism Example
ERHS Math Geometry
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
4
3
5
Mr. Chin-Sung Lin
79
Pyramids
ERHS Math Geometry
Mr. Chin-Sung Lin
80
Pyramids
ERHS Math Geometry
A pyramid is a solid figure with a base that is a
polygon and lateral faces that are triangles
Mr. Chin-Sung Lin
81
Pyramids Vertex Altitude
ERHS Math Geometry
Vertex All lateral edges meet in a
point Altitude the perpendicular line segment
from the vertex to thebase
Vertex
Vertex
Altitude
Altitude
Mr. Chin-Sung Lin
82
Regular Pyramids
ERHS Math Geometry
Slant Height
Altitude
A 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
83
Surface Area of Pyramids
ERHS Math Geometry
Slant Height
The 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
84
Volume of Pyramids
ERHS Math Geometry
The 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
Height
Base Area
Mr. Chin-Sung Lin
85
Volume of Pyramids Example
ERHS Math Geometry

• A 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
• What is the total surface area of the pyramid?
• What is the volume of the pyramid?

13
10
Mr. Chin-Sung Lin
86
Volume of Pyramids Example
ERHS Math Geometry

• A 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
• What is the total surface area of the pyramid?
• What is the volume of the pyramid?

13
10
Mr. Chin-Sung Lin
87
Volume of Pyramids Example
ERHS Math Geometry

• A 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
• What is the total surface area of the pyramid?
• What is the volume of the pyramid?

12
13
5
10
Mr. Chin-Sung Lin
88
Volume of Pyramids Example
ERHS Math Geometry
a. 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
12
13
5
10
Mr. Chin-Sung Lin
89
Properties of Regular Pyramids
ERHS Math Geometry
The 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
90
Cylinders
ERHS Math Geometry
Mr. Chin-Sung Lin
91
Cylinders
ERHS Math Geometry
The solid figure formed by the congruent parallel
curves and the surface that joins them is called
a cylinder
Mr. Chin-Sung Lin
92
Cylinders
ERHS Math Geometry
Altitude
Bases 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
Lateral Surface
Bases
Mr. Chin-Sung Lin
93
Circular Cylinders
ERHS Math Geometry
A cylinder whose bases are congruent circles
Mr. Chin-Sung Lin
94
Right Circular Cylinders
ERHS Math Geometry
If 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
95
Surface Area of Right Circular Cylinders
ERHS Math Geometry
Base Area 2pr2 Lateral Area 2prh Total Surface
Area 2prh 2pr2
Mr. Chin-Sung Lin
96
Volume of Circular Cylinders
ERHS Math Geometry
Volume B x h pr2h
Mr. Chin-Sung Lin
97
Right Circular Cylinders Example
ERHS Math Geometry
A right cylinder as shown in the
diagram. Calculate the total Surface Area
Calculate the volume
Mr. Chin-Sung Lin
98
Right Circular Cylinders Example
ERHS Math Geometry
Base Area 2pr2 2p62 226.19 Lateral Area
2prh 2p (6)(14) 527.79 Total Surface Area
226.19 527.79 754.58 Volume B x h pr2h
p(62)(14) 1583.36
Mr. Chin-Sung Lin
99
Cones
ERHS Math Geometry
Mr. Chin-Sung Lin
100
Right Circular Conical Surface
ERHS Math Geometry
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
101
Right Circular Cone
A
ERHS Math Geometry
The 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
102
Surface Area of a Cone
A
ERHS Math Geometry
Base Area B pr2 Lateral Area L ½ Chs ½
(2pr)hs prhs Total Surface Area prhs pr2
hs slant height hc height r radius
B base area C circumference
hc
B
Mr. Chin-Sung Lin
103
Volume of a Cone
A
ERHS Math Geometry
Base Area B pr2 Volume V ? Bhc ? pr2hc
hs slant height hc height r radius
B base area C circumference
hc
C
Mr. Chin-Sung Lin
104
Surface Area of a Cone Example
A
ERHS Math Geometry
Calculate the base area, lateral area, and total
area
24
Mr. Chin-Sung Lin
105
Surface Area of a Cone Example
A
ERHS Math Geometry
Calculate the base area, lateral area, and total
area Base Area B p(10)2 100p Lateral Area L
p(10)(26) 260p Total Surface Area 100p
260p 360p
24
Mr. Chin-Sung Lin
106
Volume of a Cone Example
A
ERHS Math Geometry
A 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
pr2h Volume of Cone V ? p32h 3ph pr2h 3ph,
r2 3, r v3 cm
h
Mr. Chin-Sung Lin
107
Spheres
ERHS Math Geometry
Mr. Chin-Sung Lin
108
Spheres
ERHS Math Geometry
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
O
r
Mr. Chin-Sung Lin
109
Sphere and Plane
ERHS Math Geometry
If the distance of a plane from the center of a
sphere is d and the radius of the sphere is r
r lt d no points in common
r d one points in common
r gt d infinite points in common (circle)
r
r
d
d
O
r
d
P
p
p
p
Mr. Chin-Sung Lin
110
Circles
ERHS Math Geometry
A circle is the set of all points in a plane
equidistant from a fixed point in the plane
called the center
r
O
p
Mr. Chin-Sung Lin
111
Theorem about Circles
ERHS Math Geometry
The 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
O
r
p
r
Mr. Chin-Sung Lin
112
Great Circle of a Sphere
ERHS Math Geometry
A great circle of a sphere is the intersection of
a sphere and a plane through the center of the
sphere
O
r
p
r
Mr. Chin-Sung Lin
113
Theorem about Circles
ERHS Math Geometry
If 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
114
Theorem about Circles
ERHS Math Geometry
If 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
115
Theorem about Circles
ERHS Math Geometry
Statements Reasons 1. Draw a line OC, point
C on plane p 1. Given, create two triangles
OC?AC, OC?BC 2. ?OCA and ?OCB
are right angles 2. Definition of
perpendicular 3. OA ? OB 3. Radius of a
sphere 4. OC ? OC 4. Reflexive postulate 5.
?OAC ? ?OBC 5. HL postulate 6. CA ? CB 6.
CPCTC 7. The intersection is a circle 7.
Definition of circles
Mr. Chin-Sung Lin
116
Theorem about Circles
ERHS Math Geometry
The intersection of a plane and a sphere is a
circle A great circle is the largest circle that
can be drawn on a sphere
p
Mr. Chin-Sung Lin
117
Theorem about Circles
ERHS Math Geometry
If two planes are equidistant from the center of
a sphere and intersect the sphere, then the
intersections are congruent circles
Mr. Chin-Sung Lin
118
Surface Area of a Sphere
A
ERHS Math Geometry
Surface Area S 4pr2 r radius
Mr. Chin-Sung Lin
119
Volume of a Sphere
A
ERHS Math Geometry
Volume V 4/3 pr3 r radius
Mr. Chin-Sung Lin
120
Sphere Example
A
ERHS Math Geometry
Find the surface area and the volume of a sphere
whose radius is 6 cm
Mr. Chin-Sung Lin
121
Sphere Example
A
ERHS Math Geometry
Find the surface area and the volume of a sphere
whose radius is 6 cm Surface Area S 4p62
144p cm2 Volume V 4/3 p63 288p cm3
Mr. Chin-Sung Lin
122
Q A
ERHS Math Geometry
Mr. Chin-Sung Lin
123
The End
ERHS Math Geometry
Mr. Chin-Sung Lin