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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)
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