Section 9.4 Volume and Surface Area
What You Will Learn Volume Surface Area 9.4-2
Volume Volume is the measure of the capacity of a three-dimensional figure. It is the amount of material you can put inside a three-dimensional figure. Surface area is sum of the areas of the surfaces of a three-dimensional figure. It refers to the total area that is on the outside surface of the figure. 9.4-3
Volume Solid geometry is the study of threedimensional solid figures, also called space figures. Volumes of threedimensional figures are measured in cubic units such as cubic feet or cubic meters. Surface areas of threedimensional figures are measured in square units such as square feet or square meters. 9.4-4
Volume Formulas Figure Rectangular Solid Cube Cylinder Cone Sphere Formula V = lwh V V = s 3 V = πr 2 h V = = 1 3 π 4 3 2 r h π r 3 Diagram h l w s s s r h h r 9.4-5
Surface Area Formulas Figure Rectangular Solid Cube Formula SA=2lw + 2wh +2lh SA= 6s 2 Diagram l s w h s s Cylinder SA = 2πrh + 2πr 2 r h Cone SA = πr 2 + πr r 2 + h 2 r h Sphere SA = 4πr 2 r 9.4-6
9.4-7
Example 1: Volume and Surface Area Determine the volume and surface area of the following threedimensional figure. Solution V = lwh = 11 3 6 = 198 ft 3 SA = 2lw + 2wh + 2lh = 2 11 3 + 2 3 6 + 2 11 6 = 66 + 36 + 132 = 234 ft 2 9.4-8
Example 1: Volume and Surface Area Determine the volume and surface area of the following threedimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths. 9.4-9
Example 1: Volume and Surface Area Solution V = π r 2 h = π 4 2 8 = 128π 402.12 m 3 SA = 2π rh + 2π r 2 = 2π 4 8 + 2π 4 2 = 64π + 32π = 96π 301.59 m 2 9.4-10
Example 1: Volume and Surface Area Determine the volume and surface area of the following threedimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths. 9.4-11
Example 1: Volume and Surface Area Solution V = 1 π r 2 h = 1 π 32 8 3 3 = 24π 75.40 m 3 SA = π r 2 + π r r 2 + h 2 = π 3 2 + π 3 3 2 + 8 2 = 9π + 3π 73 108.80 m 2 9.4-12
Example 1: Volume and Surface Area Determine the volume and surface area of the following three-dimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths. 9.4-13
Example 1: Volume and Surface Area Solution V = 4 π r 3 = 4 π 93 3 3 = 972π 3053.63 cm 3 SA = 4π r 2 = 4 π 9 2 = 4 π 81 = 324π 1017.88 cm 2 9.4-14
Polyhedra A polyhedron is a closed surface formed by the union of polygonal regions. 9.4-15
Euler s Polyhedron Formula Number number number of vertices of + edges of faces = 2 9.4-16
Platonic Solid A platonic solid, also known as a regular polyhedron, is a polyhedron whose faces are all regular polygons of the same size and shape. There are exactly five platonic solids. Tetrahedron: Cube: Octahedron: Dodecahedron: Icosahedron: 4 faces, 6 faces, 8 faces, 12 faces, 20 faces, 4 vertices, 6 edges 8 vertices, 12 edges 6 vertices, 12 edges 20 vertices, 30 edges 12 vertices, 30 edges 9.4-17
Prism A prism is a special type of polyhedron whose bases are congruent polygons and whose sides are parallelograms. These parallelogram regions are called the lateral faces of the prism. If all the lateral faces are rectangles, the prism is said to be a right prism. 9.4-18
Prism The prisms illustrated are all right prisms. When we use the word prism in this book, we are referring to a right prism. 9.4-19
Volume of a Prism V = Bh, where B is the area of the base and h is the height. 9.4-20
Example 6: Volume of a Hexagonal Prism Fish Tank Frank Nicolzaao s fish tank is in the shape of a hexagonal prism. Use the dimensions shown in the figure and the fact that 1 gal = 231 in 3 to a) determine the volume of the fish tank in cubic inches. 9.4-21
Example 6: Volume of a Hexagonal Prism Fish Tank Solution Area of hexagonal base: two identical trapezoids A = 1 h ( b + b ) trap 2 1 2 A trap = 1 2 (8)(16 + 8) = 96 in2 Area base = 2(96) = 192 in 2 9.4-22
Example 6: Volume of a Hexagonal Prism Fish Tank Solution Volume of fish tank: V = B h = 192 24 = 4608 in 3 9.4-23
Example 6: Volume of a Hexagonal Prism Fish Tank b) determine the volume of the fish tank in gallons (round your answer to the nearest gallon). Solution V = 4608 231 19.95 gal 9.4-24
Pyramid A pyramid is a polyhedron with one base, all of whose faces intersect at a common vertex. 9.4-25
Volume of a Pyramid where B is the area of the base and h is the height. V = 1 3 Bh 9.4-26
Example 8: Volume of a Pyramid Determine the volume of the pyramid. Solution Area of base = s 2 = 2 2 = 4 m 2 V = 1 3 Bh = 1 3 4 3 = 4 m 3 The volume is 4 m 3. 9.4-27