8.4 Volume and Surface Area 493 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. All five platonic solids are illustrated in the Did You Know? to the left. 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 . The prisms illustrated in Fig. 8.40 are all right prisms. When we use the word prism in this book, we are referring to a right prism. (d) (c) (b) (a) Bottom base Top base Prisms Figure 8.40 The volume of any prism can be determined by multiplying the area of the base, B, by the height, h, of the prism. Did You Know? Platonic Solids Tetrahedron: 4 faces, 4 vertices, 6 edges Cube: 6 faces, 8 vertices, 12 edges Octahedron: 8 faces, 6 vertices, 12 edges Dodecahedron: 12 faces, 20 vertices, 30 edges Icosahedron: 20 faces, 12 vertices, 30 edges Aplatonic solid is a polyhedron whose faces are all regular polygons of the same size and shape. There are exactly five platonic solids, as shown above. Platonic solids, also called regular polyhedra, are named after the ancient Greek philosopher Plato and are included in Euclid’s Elements (see the Profile in Mathematics on page 453). The figures above show each of the platonic solids, along with the number of faces, vertices, and edges for each. Example 5 Using Euler’s Polyhedron Formula A certain polyhedron has 20 vertices and 12 faces. Determine the number of edges on the polyhedron. Solution Since we are seeking the number of edges, we will let x represent the number of edges on the polyhedron. Next, we will use Euler’s polyhedron formula to set up an equation: − + = − + = − = − = − = x x x x Number of vertices number of edges number of faces 2 20 12 2 32 2 30 30 Therefore, the polyhedron has 30 edges. 7 Now try Exercise 35 Volume of a Prism V Bh = where B is the area of a base and h is the height. Example 6 Volume of a Hexagonal Prism Fish Tank Frank’s fish tank is in the shape of a hexagonal prism, as shown in Fig. 8.41. Use the dimensions shown in the figure and the fact that 1 gal 231 in.3 = to determine the volume of the fish tank in a) cubic inches. b) gallons (round your answer to the nearest gallon).
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