2 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.
3 Polyhedrons Below is a rectangular prism, which is a polyhedron. A B Specific Name of Solid: Rectangular PrismD 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, DHVertices: A, B, C, D, E, F, G, H
4 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 BCH G CG and DC, DH and DC, DH and AD
5 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 AEH G
6 Formulas in Sect. 6.3 and Sect. 6.4 Diagonal of a Right Rectangular Prismdiagonal = √(l² + w² + h²). l = length, w = width, h = heightDistance Formula in Three Dimensionsd = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]Midpoint Formula in Three Dimensionsx₁ + x₂ , y₁ + y₂ , z₁ + z₂
7 Surface Area and Volume Section 7.1Surface Area and Volume
8 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.
9 Surface Area and Volume Rectangular PrismCubeSurface AreaS = 2ℓw + 2wh + 2ℓhVolumeV = ℓwhℓ = lengthw = widthh = heightSurface AreaS = 6s²VolumeV = s³S = Surface AreaV = Volumes = side (edge)
10 Surface Area and Volume of Prisms Section 7.2Surface Area and Volume of Prisms
11 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.
12 Surface Area of a Right Prism S = L + 2B or S = Ph + 2BS = surface area, L = Lateral Area,B = Base Area, P = Perimeter of the base,h = heightThe surface area of a prism may be broken down into two parts: the area of the bases and the area of the lateral faces.
13 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 = 2012S = Ph + 2BE F S = (18)(12) + 2(20)4 S =H G S = 256 un²
14 Volume of a PrismThe 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
15 Surface Area of a Right Prism Below is a rectangular prism, which is a polyhedron.A B B = (5)(4)D C B = 2012V = BhE F V = (20)(12)4 V = 240 un³H G
16 Surface Area and Volume of Pyramids Section 7.3Surface Area and Volume of Pyramids
17 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.
18 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.
19 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, CBF Lateral Edges: AB, AC, AD, AFB Base: BFDCLateral Faces: ∆ABC, ∆ACD, ∆ADF, ∆AFBD The yellow line is the slant height.C The green line is the height of the pyramid.
20 Surface Area of a Regular Pyramid S = L + B or S = ½ℓP + B.S = Surface Area L = Lateral Area B = Base Areaℓ = slant height P =ℓ = 10 P = 42 unitsB = (9)(12)B = 108 un²S = ½ (10)(42) + 10812 S =S = 318 un²
21 Volume of a Regular Pyramid V = ⅓ BhV = Volume B = Base Area h = height of pyramidh = 8 B = (9)(12)B = 108 un²V = ⅓ (108)(8)V = 288 un³12
22 Surface Area and Volume of Cylinders Section 7.4Surface Area and Volume of Cylinders
23 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.
24 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
25 Surface Area of a Right Cylinder S = L +2B or S = 2πrh + 2πr²S = 2π(4)(9) + 2π4²S = 2π(36) + 2π(16)9 S = 72π + 32πS = un² S = 104π un²4 (approximate answer) ( exact answer)
26 Volume of a CylinderThe volume, V, of a cylinder with radius r, height h, and base area B is:V = Bh or V = πr²h
27 Surface Area of a Right Cylinder V = Bh or V = πr²hV = π(4²)(9)V = π(16)(9)9 V = π(144)V = un³ V = 144π un³4 (approximate answer) ( exact answer)
28 Surface Area and Volume of Cones Section 7.5Surface Area and Volume of Cones