PROPOSITION XVIII. THEOREM 563. The frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the two bases of the frustum. D B Given the frustum of a triangular pyramid, ABC-DEF, having ABC, or b, for its lower base; DEF, or b', for its upper base; and the altitude a. To prove that ABC-DEF = ab + } ab' + § a √bb'. Proof. Through A, E, and C, and also through C, D, and E, pass planes dividing the frustum into three pyramids. Then and E-ABC = } ab, C-DEF = } ab'. § 559 fa√bb'. It therefore remains only to prove that E-ACD = We see by the figure that we may speak of E-ABC as C-ABE, and of E-ACD as C-AED. But C-ABE: C-AED=▲ ABE: ▲ AED. § 562 Since A ABE and AED have for a common altitude the altitude of the trapezoid ABED, or .. A ABE: ▲ AED = AB: DE. § 327 Ax. 8 Ax. 9 In like manner E-ACD and E-CFD have a common vertex E and have their bases in the same plane, ACFD, so that E-ACD: E-CFD =AACD: ▲ CFD. § 562 Since AACD and CFD have for a common altitude the altitude of the trapezoid ACFD, Ax. 8 .. E-ABC: E-ACD= E-ACD: E-CFD. But E-CFD is the same as C-DEF, which has been shown to equal ab'. .. ab: E-ACD = E-ACD : } ab'. Ax. 9 .. E-ACD= √‡ ab × § ab' $ 262 Ax.1 ... E-ABC +C-DEF+E-ACD=& ab + § ab' + § a√bb'. That is, ABC-DEF = § ab + } ab' + § a √bb', by Ax. 9. Q.E.D. 564. COROLLARY 1. The volume of a frustum of a triangular pyramid may be expressed as ‡ a (b+b' + √bb'). For we may factor by } a. 565. COROLLARY 2. The volume of a frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum. Extend the faces of the frustum F, forming a pyramid P. From a triangular pyramid P' of equivalent base b and equal altitude, cut off a frustum F' of the same altitude a as F. Then P P' and FF'. But F and F' have equivalent bases, and F' = }a(b+b′+√bb ́). Hence Fa(b+b+√bb'). = α 566. Polyhedrons classified as to Faces. A polyhedron of four faces is called a tetrahedron; one of six faces, a hexahedron; one of eight faces, an octahedron; one of twelve faces, a dodecahedron; one of twenty faces, an icosahedron. 567. Regular Polyhedron. A polyhedron whose faces are congruent regular polygons, and whose polyhedral angles are equal, is called a regular polyhedron. It is proved on page 351 that it is possible to have only five regular polyhedrons. They may be constructed from paper as follows: + Draw on stiff paper the diagrams given above. Cut through the full lines and paste strips of paper on the edges as shown. Fold on the dotted lines, and keep the edges in contact by the pasted strips of paper. PROPOSITION XIX. PROBLEM 568. To determine the number of regular convex polyhedrons possible. A convex polyhedral angle must have at least three faces, and the sum of its face angles must be less than 360° (§ 495). 1. Since each angle of an equilateral triangle is 60°, convex polyhedral angles may be formed by combining three, four, or five equilateral triangles. The sum of six such angles is 360°, and therefore is greater than the sum of the face angles of a convex polyhedral angle. Hence three regular convex polyhedrons are possible with equilateral triangles for faces. 2. Since each angle of a square is 90°, a convex polyhedral angle may be formed by combining three squares. The sum of four such angles is 360°, and therefore is greater than the sum of the face angles of a convex polyhedral angle. Hence one regular convex polyhedron is possible with squares. 3. Since each angle of a regular pentagon is 108° (§ 145), a convex polyhedral angle may be formed by combining three regular pentagons. The sum of four such angles is 432°, and therefore is greater than the sum of the face angles of a convex polyhedral angle. Hence one regular convex polyhedron is possible with regular pentagons. 4. The sum of three angles of a regular hexagon is 360°, of a regular heptagon is greater than 360°, and so on. Hence only five regular convex polyhedrons are possible. The regular polyhedrons are the regular tetrahedron, the regular hexahedron, or cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron. Q. E. F. It adds greatly to a clear understanding of the five regular polyhedrons if they are constructed from paper as suggested in § 567. Since these solids were extensively studied by the pupils of Plato, the great Greek philosopher, they are often called the Platonic Bodies. EXERCISE 92 Find the volumes of frustums of pyramids, the altitudes and the bases of the frustums being given, as follows: 1. a = 3 in., b = 8 sq. in., b' = 2 sq. in. 2. a = 41⁄2 in., b = 83 sq. in., b' = 3 sq. in. 3. α = 3.2 in., b = : 2 sq. in., b' = 0.18 sq. in. 4. a= : 2 ft. 6 in., b = 10 sq. ft., b' =2 sq. ft. 72 sq. in. 5. a 3 ft. 7 in., b = 24 sq. ft. 72 sq. in., = b' = 2 sq. ft. 6. A pyramid 2 in. high, with a base whose area is 8 sq. in., is cut by a plane parallel to the base 1 in. from the vertex. Find the volume of the frustum. 7. A pyramid 3 in. high, with a base whose area is 81 sq. in., is cut by a plane parallel to the base 2 in. from the base. Find the volume of the frustum. 8. The lower base of a frustum of a pyramid is a square 4 in. on a side. The side of the upper base is half that of the lower base, and the altitude of the frustum is the same as the side of the upper base. Find the volume of the frustum. 9. The lower base of a frustum of a pyramid is a square 3 in. on a side. The area of the upper base is half that of the lower base, and the altitude of the frustum is 2 in. Find to two decimal places the volume of the frustum. 10. A pyramid has six edges, each 1 in. long. Find to two decimal places the volume of the pyramid. 11. A regular tetrahedron has a volume 2 √2 cu. in. Find to two decimal places the length of an edge. 12. The base of a regular pyramid is a square 7 ft. on a side. The slant height is s ft. Find the area of the entire surface. 13. Consider the formula va (b+b'+ √bb'), of § 564, when b' = 0. Discuss the meaning of the result. Also discuss the case in which b=b', |