Eleanor Roosevelt High School
Chin-Sung Lin
ERHS Math Geometry
The geometry of three dimensions is called
solid geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
There is one and only one plane containing
three non-collinear points
A
Mr. Chin-Sung Lin
ERHS Math Geometry
A plane containing any two points contains all
of the points on the line determined by
those two points
A
Mr. Chin-Sung Lin
ERHS Math Geometry
There is exactly one plane containing a line and
a point not on the line
A
Mr. Chin-Sung Lin
ERHS Math Geometry
If two lines intersect, then there is exactly one
plane containing them
Two intersecting lines determine a plane
A
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Skew lines are lines in space that are neither
parallel nor intersecting
Mr. Chin-Sung Lin
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
E
A
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
If two planes intersect, then they intersect in
exactly one line
A
Mr. Chin-Sung Lin
ERHS Math Geometry
A dihedral angle is the union of two half-planes
with a common edge
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular planes are two planes that intersect to
form a right dihedral angle
AC
AC
AB, AD AB, and
AD (m CAD = 90)
then
m
n
A
Mr. Chin-Sung Lin
ERHS Math Geometry
If a line not in a plane intersects the plane, then
it intersects in exactly one point
A
Mr. Chin-Sung Lin
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
then k s
n,
Mr. Chin-Sung Lin
ERHS Math 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
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
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
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
Mr. Chin-Sung Lin
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
ΔRAP = ΔSAP
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
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
ΔRBP = ΔSBP
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
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
ΔRAB = ΔSAB
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
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
ΔRAQ = ΔSAQ
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
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
ΔRPQ = ΔSPQ
Mr. Chin-Sung Lin
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
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
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
Prove: p
q
q
A
Mr. Chin-Sung Lin
ERHS Math Geometry
Two planes are perpendicular if and only if one plane contains
a line perpendicular to the other
A
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
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
P
Mr. Chin-Sung Lin
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
P
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
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
A
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel planes are planes that have no points in
common
m
n
Mr. Chin-Sung Lin
ERHS Math Geometry
A line is parallel to a plane if it has no points in
common with the plane
m
Mr. Chin-Sung Lin
ERHS Math Geometry
If a plane intersects two parallel planes, then the
intersection is two parallel lines
p
m
n
Mr. Chin-Sung Lin
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
m
n
Mr. Chin-Sung Lin
ERHS Math Geometry
Two lines perpendicular to the same plane are parallel
Given: Plane p, LA⊥p at A, and MB⊥p at B
q
Prove: LA//MB
p
Mr. Chin-Sung Lin
ERHS Math Geometry
Two lines perpendicular to the same plane are parallel
Given: Plane p, LA⊥p at A, and MB⊥p at B
q
Prove: LA//MB
p
Mr. Chin-Sung Lin
ERHS Math Geometry
Two lines perpendicular to the same plane are coplanar
Given: Plane p, LA⊥p at A, and MB⊥p at B
q
Prove: LA and MB are coplanar
p
Mr. Chin-Sung Lin
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
p
q
Mr. Chin-Sung Lin
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
s
Prove: p//q
p
q
Mr. Chin-Sung Lin
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
p
Prove: q⊥AB
q
Mr. Chin-Sung Lin
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
p
Prove: q⊥AB
q
Mr. Chin-Sung Lin
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
p
Prove: q⊥AB
q
Mr. Chin-Sung Lin
ERHS Math Geometry
Two planes are perpendicular to the same line if and
only if the planes are parallel
p
q
Mr. Chin-Sung Lin
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
p
q
Mr. Chin-Sung Lin
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
p
q
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
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
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
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
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
ERHS Math Geometry
The lateral edges of a prism are congruent and parallel
Lateral Edges
Mr. Chin-Sung Lin
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
ERHS Math Geometry
A parallelepiped is a prism that has parallelograms as
bases
Mr. Chin-Sung Lin
ERHS Math Geometry
A rectangular parallelepiped is a parallelepiped that has
rectangular bases and lateral edges perpendicular to
the bases
Mr. Chin-Sung Lin
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
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
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
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
4
5
7
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Area of the bases:
½ x (4 x 2√3) x 2= 8√3
Lateral area:
3 x (4 x 5) = 60
4
2√3
Total surface area: 60 + 8√3 ≈ 73.86
5
2
4
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
The volume (V) of a prism is equal to the area of the base
(B) times the height (h)
V=Bxh
Height (h)
Base (B)
Mr. Chin-Sung Lin
ERHS Math Geometry
A right prism is shown in the diagram
2
Calculate the Volume of the prism
5
4
Mr. Chin-Sung Lin
ERHS Math Geometry
A right prism is shown in the diagram
2
Calculate the Volume of the prism
B=½x4x2=4
h=5
5
V = Bh = 4 x 5 = 20
4
Mr. Chin-Sung Lin
ERHS Math Geometry
A right prism is shown in the diagram
Calculate the Volume of the prism
4
3
5
Mr. Chin-Sung Lin
ERHS Math Geometry
A right prism is shown in the diagram
Calculate the Volume of the prism
4
B = 5 x 4 = 20
h=3
3
V = Bh = 20 x 3 = 60
5
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Vertex: All lateral edges meet in a point
Altitude: the perpendicular line segment from the vertex
to thebase
Vertex
Altitude
Vertex
Altitude
Mr. Chin-Sung Lin
ERHS Math Geometry
Slant Height
A pyramid whose base is a regular polygon and
whose altitude is perpendicular to the base
at its center
Altitud
e
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 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
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)
Height
V = (1/3) x B x h
Base Area
Mr. Chin-Sung Lin
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
a.
What is the total surface area of the
pyramid?
b.
What is the volume of the pyramid?
13
10
Mr. Chin-Sung Lin
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
a.
What is the total surface area of the
pyramid?
b.
What is the volume of the pyramid?
13
10
Mr. Chin-Sung Lin
ERHS Math Geometry
A regular pyramid has a square base. The
12
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?
13
5
10
Mr. Chin-Sung Lin
ERHS Math Geometry
a. Total surface area:
Lateral Area: ½ x 10 x 13 x 4 = 260
12
13
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
5
10
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Mr. Chin-Sung Lin
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
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
Lateral
Bases
Surface
Height: the length of an altitude
Mr. Chin-Sung Lin
ERHS Math Geometry
A cylinder whose bases are congruent circles
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Base Area: 2πr2
r
Lateral Area: 2πrh
Total Surface Area: 2πrh + 2πr2
h
Mr. Chin-Sung Lin
ERHS Math Geometry
Volume: B x h = πr2h
Mr. Chin-Sung Lin
ERHS Math Geometry
A right cylinder as shown in the diagram.
6
Calculate the total Surface Area
Calculate the volume
14
Mr. Chin-Sung Lin
ERHS Math Geometry
Base Area:
6
2πr2 = 2π62 ≈ 226.19
Lateral Area:
2πrh = 2π (6)(14) ≈ 527.79
14
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
Mr. Chin-Sung Lin
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
QA
circular conical surface
* A conical surface extends infinitely
p
O
P
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
The part of the conical surface generated
by PQ from plane p to Q is called a
right circular cone
QA
Q: vertex of the cone
Circle O: base of the cone
OQ: altitude of the cone
OQ: height of the cone, and
p
O
P
C
PQ: slant height of the cone
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Base Area: B = πr2
Lateral Area: L = ½ Chs= ½ (2πr)hs = πrhs
A
Total Surface Area: πrhs + πr2
*
*
*
*
*
hs:
hc:
r:
B:
C:
slant height
height
radius
base area
circumference
hs
hc
p
r
B
C
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Base Area: B = πr2
A
Volume: V = ⅓ Bhc= ⅓ πr2hc
*
*
*
*
*
hs:
hc:
r:
B:
C:
slant height
height
radius
base area
circumference
hs
hc
p
r
B
C
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Calculate the base area, lateral area, and
total area
A
24
26
10
p
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Calculate the base area, lateral area, and
total area
A
Base Area: B = π(10)2 = 100π
24
Lateral Area: L = π(10)(26) = 260π
Total Surface Area: 100π + 260π
= 360π
26
10
p
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
r
A cone and a cylinder have equal volumes
h
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?
A
Volume of Cylinder: V = h = πr2h
h
Volume of Cone: V = ⅓ π32h = 3πh
πr2h = 3πh, r2 = 3, r = √3 cm
3 cm
p
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
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
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 < d no points
in common
r = d one points
in common
O
r
p
r
d
P
p
O
P
r > d infinite points
in common (circle)
d
rO
p
d
P
Mr. Chin-Sung Lin
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
p
O
r
Mr. Chin-Sung Lin
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
p
r
r
Mr. Chin-Sung Lin
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
p
r
r
Mr. Chin-Sung Lin
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
O
Prove: The intersection is a circle
p
A
C
B
Mr. Chin-Sung Lin
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
O
Prove: The intersection is a circle
p
A
C
r
B
Mr. Chin-Sung Lin
ERHS Math Geometry
O
A
p
Statements
C
r
B
Reasons
1. Draw a line OC, point C on plane p
OCAC, OCBC
1. Given, create two triangles
2. OCA and
3. OA  OB
2. Definition of perpendicular
3. Radius of a sphere
OCB are right angles
4. OC  OC
5. OAC  OBC
6. CA  CB
4. Reflexive postulate
5. HL postulate
6. CPCTC
7. The intersection is a circle
7. Definition of circles
Mr. Chin-Sung Lin
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
O
p’
p
Mr. Chin-Sung Lin
ERHS Math Geometry
If two planes are equidistant from the center of a sphere and
intersect the sphere, then the intersections are congruent
circles
C
p
O
D
q
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
A
Surface Area: S = 4πr2
r:
radius
O
r
Mr. Chin-Sung Lin
ERHS Math Geometry
A
Volume: V = 4/3 πr3
r:
radius
O
r
Mr. Chin-Sung Lin
ERHS Math Geometry
A
Find the surface area and the volume of a
sphere whose radius is 6 cm
O
r
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Find the surface area and the volume of a
sphere whose radius is 6 cm
Surface Area: S =
4π62
= 144π
cm2
O
r
Volume: V = 4/3 π63 = 288π cm3
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin
ERHS Math Geometry
Mr. Chin-Sung Lin