Section 6.2
Spatial Relationships
Figures in Space
• Closed spatial figures are known as solids.
• A polyhedron is a closed spatial figure
composed of polygons, called the faces of the
polyhedron.
• The intersections of the faces are the edges of
the polyhedron.
• The vertices of the faces are the vertices of
the polyhedron.
Polyhedrons
• Below is a rectangular prism, which is a polyhedron.
A
B
D
C
E
H
F
G
Specific Name of Solid: Rectangular Prism
Name of Faces: ABCD (Top),
EFGH (Bottom),
DCGH (Front),
ABFE (Back),
AEHD (Left),
CBFG (Right)
Name of Edges: AB, BC, CD, DA, EF, FG,
GH, HE, AE, BF, CG, DH
Vertices: A, B, C, D, E, F, G, H
Intersecting, Parallel, and Skew Lines
• Below is a rectangular prism, which is a polyhedron.
A
B
D
C
E
H
F
G
Intersecting Lines: AB and BC, BC and CD,
CD and DA, DA and AB, AE and EF,
AE and EH, BF and EF, BF and FG,
CG and FG, CG and GH, DH and GH,
DH and EH, AE and DA, AE and AB,
BF and AB, BF and BC, CG and BC
CG and DC, DH and DC, DH and AD
Intersecting, Parallel, and Skew Lines
• Below is a rectangular prism, which is a polyhedron.
A
D
Parallel Lines: AB, DC, EF, and HG;
AD, BC, EH, and FG;
AE, BF, CG, and DH.
F
Skew Lines: (Some Examples)
AB and CG, EH and BF, DC and AE
C
E
H
B
G
Formulas in Sect. 6.3 and Sect. 6.4
•
•
•
•
•
•
Diagonal of a Right Rectangular Prism
diagonal = √(l² + w² + h²). l = length, w = width, h = height
Distance Formula in Three Dimensions
d = √*(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²+
Midpoint Formula in Three Dimensions
x₁ + x₂ , y₁ + y₂ , z₁ + z₂
2
2
2
Section 7.1
Surface Area and Volume
Surface Area and Volume
• The surface area of an
object is the total area
of all the exposed
surfaces of the object.
• The volume of a solid
object is the number of
nonoverlapping unit
cubes that will exactly
fill the interior of the
figure.
Surface Area and Volume
Rectangular Prism
• Surface Area
• S = 2ℓw + 2wh + 2ℓh
Cube
• Surface Area
• S = 6s²
• Volume
• V = ℓwh
• Volume
• V = s³
• ℓ = length
• w = width
• h = height
• S = Surface Area
• V = Volume
• s = side (edge)
Section 7.2
Surface Area and Volume of Prisms
Surface Area of Right Prisms
• An altitude of a prism is a segment that has
endpoints in the planes containing the bases
and that is perpendicular to both planes.
• The height of a prism is the length of an
altitude.
Surface Area of a Right Prism
•
•
•
•
•
S = L + 2B or S = Ph + 2B
S = surface area, L = Lateral Area,
B = Base Area, P = Perimeter of the base,
h = height
The surface area of a prism may be broken
down into two parts: the area of the bases
and the area of the lateral faces.
Surface Area of a Right Prism
• Below is a rectangular prism, which is a polyhedron.
A
B
D
C
P = 5 + 4 + 5 + 4 B = (5)(4)
P = 18
B = 20
12
E
H
5
F
4
G
S = Ph + 2B
S = (18)(12) + 2(20)
S = 216 + 40
S = 256 un²
Volume of a Prism
• The volume of a solid measures how much
space the solid takes or can hold.
• The volume, V, of a prism with height, h, and
base area, B is:
• V = Bh
Volume of a Right Prism
• Below is a rectangular prism, which is a polyhedron.
A
B
D
C
B = (5)(4)
B = 20
12
E
H
5
F
4
G
V = Bh
V = (20)(12)
V = 240 un³
Section 7.3
Surface Area and Volume of Pyramids
Properties of Pyramids
• A pyramid is a polyhedron consisting of one
base, which is a polygon, and three or more
lateral faces.
• The lateral faces are triangles that share a
single vertex, called the vertex of the pyramid.
• Each lateral face has one edge in common
with the base, called a base edge. The
intersection of two lateral faces is a lateral
edge.
Properties of Pyramids
• The altitude of a pyramid is the perpendicular
segment from the vertex to the plane of the base.
• The height of a pyramid is the length of its
altitude.
• A regular pyramid is a pyramid whose base is a
regular polygon and whose lateral faces are
congruent isosceles triangles.
• The length of an altitude of a lateral face of a
regular pyramid is called the slant height.
Surface Area of a Regular Pyramid
• S = L + B or S = ½ℓp + B.
A
F
B
D
C
A is the vertex of the pyramid.
B, F, D, and C are the other vertices.
Base Edges: BF, FD, DC, CB
Lateral Edges: AB, AC, AD, AF
Base: BFDC
Lateral Faces: ∆ABC, ∆ACD, ∆ADF, ∆AFB
The yellow line is the slant height.
The green line is the height of the
pyramid.
Surface Area of a Regular Pyramid
• S = L + B or S = ½ℓP + B.
8
10
9
12
S = Surface Area L = Lateral Area B = Base Area
ℓ = slant height P = 9 + 12 + 9 + 12
ℓ = 10
P = 42 units
B = (9)(12)
B = 108 un²
S = ½ (10)(42) + 108
S = 210 + 108
S = 318 un²
Volume of a Regular Pyramid
• V = ⅓ Bh
V = Volume B = Base Area h = height of pyramid
h=8
10
8
B = (9)(12)
B = 108 un²
V = ⅓ (108)(8)
V = 288 un³
9
12
Section 7.4
Surface Area and Volume of Cylinders
Properties of Cylinders
• A cylinder is a solid that consists of a circular region and its
translated image on a parallel plane, with a lateral surface
connecting the circles.
• The bases of a cylinder are circles.
• An altitude of a cylinder is a segment that has endpoints in
the planes containing the bases and is perpendicular to
both bases.
• The height of a cylinder is the length of the altitude.
• The axis of a cylinder is the segment joining the centers of
the two bases.
• If the axis of a cylinder is perpendicular to the bases, then
the cylinder is a right cylinder. If not, it is an oblique
cylinder.
The Surface Area of a Right Cylinder
• The surface area, S, of a right cylinder with
lateral area L, base area B, radius r, and
height h is:
• S = L +2B or S = 2πrh + 2πr²
Surface Area of a Right Cylinder
• S = L +2B or S = 2πrh + 2πr²
9
4
S = 2π(4)(9) + 2π4²
S = 2π(36) + 2π(16)
S = 72π + 32π
S = 326.73 un² S = 104π un²
(approximate answer)
( exact answer)
Volume of a Cylinder
• The volume, V, of a cylinder with radius r,
height h, and base area B is:
• V = Bh or V = πr²h
Volume of a Right Cylinder
• V = Bh or V = πr²h
9
4
V = π(4²)(9)
V = π(16)(9)
V = π(144)
V = 452.39 un³
(approximate answer)
V = 144π un³
( exact answer)
Section 7.5
Surface Area and Volume of Cones
Properties of Cones
• A cone is a tree-dimensional figure that consists
of a circular base and a curved lateral surface that
connects the base to a single point not in the
plane of the base, called the vertex.
• The altitude of a cone is the perpendicular
segment from the vertex to the plane of the base.
• The height of the cone is the length of the
altitude.
• If the altitude of a cone intersects the base of the
cone at its center, the cone is a right cone. If not,
it is an oblique cone.
Surface Area of a Right Cone
• The surface area, S, or a right cone with lateral
area L, base of area B, radius r, and slant
height ℓ is:
• S = L + B or S = πrℓ + πr²
Surface Area of a Right Cone
• S = L + B or S = πrℓ + πr²
8
10
6
S = π(6)(10) + π(6²)
S = 60π + 36π
S = 96π
S = 301.59 units² S = 96π units²
(approximate answer) (exact answer)
Volume of a Cone
• The volume, V, of a cone with radius r, height
h, and base area B is:
• V = ⅓Bh or V = ⅓πr²h
Volume of a Cone
• V = ⅓Bh or
8
V = ⅓πr²h
10
6
V = ⅓π(6²)(8)
V = ⅓π(36)(8)
V = ⅓π(288)
V = 96π
S = 301.59 units³ V = 96π units³
(approximate answer) (exact answer)
Section 7.6
Surface Area and Volume of Spheres
Properties of Spheres
• A sphere is the set of all points in space that
are the same distance, r, from a given point
known as the center of the sphere.
Surface Area of a Sphere
• The surface area, S, of a sphere with radius r
is:
• S = 4πr²
Surface Area of a Sphere
• S = 4πr²
S = 4π(7²)
S = 4π(49)
S = 196π
7
S = 615.75 units² S = 196π units²
(approximate answer) (exact answer)
Volume of a Sphere
• The volume, V, of a sphere with radius r is:
• V = ⁴⁄₃πr³
Volume of a Sphere
• V = ⁴⁄₃πr³
V = ´⁄₃π(7³)
V = ´⁄₃π(343)
V = ¹³⁷²⁄₃ π
7
V = 1436.76 units³ V = ¹³⁷²⁄₃ π units³
(approximate answer)
(exact answer)