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 Specific Name of Solid: Rectangular Prism

D C Name of Faces: ABCD (Top),

EFGH (Bottom),

DCGH (Front),

E F ABFE (Back),

AEHD (Left),

H G 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 Intersecting Lines: AB and BC, BC and CD,

D C 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,

E F DH and EH, AE and DA, AE and AB,

BF and AB, BF and BC, CG and BC

H G CG and DC, DH and DC, DH and AD

Intersecting, Parallel, and Skew Lines

• Below is a rectangular prism, which is a polyhedron.

A B Parallel Lines: AB, DC, EF, and HG;

D C AD, BC, EH, and FG;

AE, BF, CG, and DH.

E F Skew Lines: (Some Examples)

AB and CG, EH and BF, DC and AE

H 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

• Volume

• V = ℓwh

• ℓ = length

• w = width

• h = height

Cube

• Surface Area

• S = 6s²

• Volume

• V = s³

• 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 P = 5 + 4 + 5 + 4 B = (5)(4)

D C P = 18 B = 20

12

S = Ph + 2B

E F S = (18)(12) + 2(20)

4 S = 216 + 40

H 5 G 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 B = (5)(4)

D C B = 20

12

V = Bh

E F V = (20)(12)

4 V = 240 un³

H 5 G

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 A is the vertex of the pyramid.

B, F, D, and C are the other vertices.

Base Edges: BF, FD, DC, CB

F Lateral Edges: AB, AC, AD, AF

B Base: BFDC

Lateral Faces: ∆ABC, ∆ACD, ∆ADF, ∆AFB

D The yellow line is the slant height.

C The green line is the height of the pyramid.

Surface Area of a Regular Pyramid

• S = L + B or S = ½ℓP + B.S = Surface Area L = Lateral Area B = Base Area

ℓ = slant height P = 9 + 12 + 9 + 12

ℓ = 10 P = 42 units

8 B = (9)(12)

10 B = 108 un²

9 S = ½ (10)(42) + 108

12 S = 210 + 108

S = 318 un²

Volume of a Regular Pyramid

• V = ⅓ BhV = Volume B = Base Area h = height of pyramid

h = 8 B = (9)(12)

10 8 B = 108 un²

V = ⅓ (108)(8)

9 V = 288 un³

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

S = 2π(4)(9) + 2π4²

S = 2π(36) + 2π(16)

9 S = 72π + 32π

S = 326.73 un² S = 104π un²

4 (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

V = π(4²)(9)

V = π(16)(9)

9 V = π(144)

V = 452.39 un³ V = 144π un³

4 (approximate answer) ( 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

S = π(6)(10) + π(6²)

S = 60π + 36π

8 10 S = 96π

S = 301.59 units² S = 96π units²

6 (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 V = ⅓πr²h

V = ⅓π(6²)(8)

V = ⅓π(36)(8)

8 10 V = ⅓π(288)

V = 96π

S = 301.59 units³ V = 96π units³

6 (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 = 4πr²S = 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 π7

V = 1436.76 units³ V = ¹³⁷²⁄₃ π units³(approximate answer) (exact answer)