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**Spatial Relationships**

Section 6.2 Spatial Relationships

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**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.

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**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

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**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

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**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

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**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

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**Surface Area and Volume**

Section 7.1 Surface Area and Volume

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**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.

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**Surface Area and Volume**

Rectangular Prism Cube Surface Area S = 2ℓw + 2wh + 2ℓh Volume V = ℓwh ℓ = length w = width h = height Surface Area S = 6s² Volume V = s³ S = Surface Area V = Volume s = side (edge)

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**Surface Area and Volume of Prisms**

Section 7.2 Surface Area and Volume of Prisms

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**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.

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**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.

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**Surface Area of a Right Prism**

Below is a rectangular prism, which is a polyhedron. A B P = B = (5)(4) D C P = 18 B = 20 12 S = Ph + 2B E F S = (18)(12) + 2(20) 4 S = H G S = 256 un²

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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

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**Surface Area 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 G

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**Surface Area and Volume of Pyramids**

Section 7.3 Surface Area and Volume of Pyramids

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**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.

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**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.

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**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.

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**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 = ℓ = 10 P = 42 units B = (9)(12) B = 108 un² S = ½ (10)(42) + 108 12 S = S = 318 un²

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**Volume of a Regular Pyramid**

V = ⅓ Bh V = Volume B = Base Area h = height of pyramid h = 8 B = (9)(12) B = 108 un² V = ⅓ (108)(8) V = 288 un³ 12

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**Surface Area and Volume of Cylinders**

Section 7.4 Surface Area and Volume of Cylinders

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**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.

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**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

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**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 = un² S = 104π un² 4 (approximate answer) ( exact answer)

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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

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**Surface Area of a Right Cylinder**

V = Bh or V = πr²h V = π(4²)(9) V = π(16)(9) 9 V = π(144) V = un³ V = 144π un³ 4 (approximate answer) ( exact answer)

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**Surface Area and Volume of Cones**

Section 7.5 Surface Area and Volume of Cones

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