Then in P-XQR V= įh · PQR, and in P-QDR V,= hPQR. $123 The pyramids P-QCD and P-QDR have the same vertex P, but, since the base CD of AQCD is twice the base QR of AQDR ($9, 17), and these A have the same altitude ($ 249), the base QCD is twice the base QDR ($ 26, 1). Hence the pyramid P-QCD is equivalent to twice the pyramid P-QDR; or, in pyramid P-QCD, V = 8 • PQR. Ax. 3 Hence in pyramid P-XCD, which is composed of P-XQR, P-QDR, and P-QCD, Vp=4 hPQR. Ax. 1 Similarly, the volume of each lateral pyramid is & h times the area of that part of the midsection included within it; and hence for the sum of all the lateral pyramids, Vp=4 h.M. Ax. 1 The volume of the pyramid with base CDE... is hB, and that of the pyramid with base XY... is į hB'. $ 123 V=h(B+B'+4 M). Ax. 1 Exercises. The Prismoid Deduce the following formulas as cases of a prismoid: 1. Cube, V=e. 3. Pyramid, V=ž Bh. 2. Prism, V=Bh. 4. Parallelepiped, V= Bh. 5. Frustum of a pyramid, Vršh(B+B'+V BB'). 6. The area of the upper base of a prismoid is 5 sq. in.; of the lower base, 9 sq. in.; of the midsection, 7 sq. in.; and the altitude is 4 in. Find the volume. 7. Consider Ex, 6 when each measurement is doubled. Exercises. Review 1. Given that the base of a regular pyramid is an equilateral triangle of side s and that the slant height is l, find the altitude and the volume of the pyramid. 2. Consider Ex. 1 when the base is a square of side s. 3. Given that the base of a regular pyramid is a square of side s and that the area of each lateral face is A, find the volume of the pyramid. 4. Find the volume of a truncated right triangular prism whose base has an area of 7 sq. in., and whose lateral edges are 1) in., 13 in., and 2 in. respectively. 5. In two tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other, the edges of this trihedral angle in the first are 3 in., 4 in., and 5 in. respectively, and those of the corresponding angle of the other are 5 in., 6 in., and 7 in. respectively. Find the ratio of the volumes of the tetrahedrons. 6. How many faces are there in a crystal which has five vertices and nine edges ? 17. What part of a cube is cut off by a plane passing through the vertex B' in the upper base and the diagonal AC in the lower base? 8. Two similar polyhedrons have the edges e, and ez of the first corresponding to e' and e, of the second. Ife,=4 in., ex=7 in., and e=5.6 in., how long is e? 9. By the aid of Cavalieri's Theorem, prove $ 122. 10. A wedge has for its base a rectangle 1 inches long and w inches wide. The cutting edge is e inches long, and is parallel to the base. The distance from e to the base is d inches. Write a formula for the volume of the wedge. Apply this formula to the case of l=5, w=2, e=3, d=4. III. SPHERICAL TRIANGLES Proposition 8. Two Sides and Included Angle 251. Theorem. If two triangles on the same sphere or on equal spheres have two sides and the included angle of one equal respectively to the corresponding parts of the other, the triangles are either congruent or symmetric. Given ABC and A'B'C', two on the same sphere or on equal spheres, with AB= A'B', AC= A'C', ZA= ZA', and similarly arranged; and the A ABC and A'B'X with AB= A'B', AC=A'X, ZA= LA', and arranged in reverse order. Prove that ABC is congruent to AA'B'C', and that ABC is symmetric with respect to A A'B'X. Proof. Place AABC upon AA'B'C'. Post. 4 By a proof similar to that for the corresponding case in plane geomtry (S 7,3), show that A ABC is congruent to A A'B'C'. Let A A'B'C' be symmetric with respect to AA'B'X. Now show that A A'B'X and A'B'C' have A'B'= A'B', A’X= A'C', ZXA'B'=<C'A'B', and are arranged in reverse order ($ 204), and hence that A ABC and A'B'C' have AB=A'B', AC=A'C', ZA=ZC'A'B' (Ax.5), and are similarly arranged. Then show that A ABC and A'B'C' are congruent, as above, and hence that A ABC and A'B'X are symmetric (Ax. 5). Proposition 9. Two Angles and Included Side 252. Theorem. If two triangles on the same sphere or on equal spheres have two angles and the included side of one equal respectively to the corresponding parts of the other, the triangles are either congruent or symmetric. Given ABC and A'B'C', two A on the same sphere or on equal spheres, with ZA= ZA', ZC= ZC', AC = A'C', and similarly arranged; and the A ABC and A'B'X with ZA= LA', ZC= 2X, AC= A'X, and arranged in reverse order. Prove that AABC is congruent to AA'B'C', and that AABC is symmetric with respect to AA'B'X. Proof. Place AABC upon AA'B'C'. Post. 4 By a proof similar to that for the corresponding case in plane geometry (S 7, 4), show that A ABC is congruent to A A'B'C'. Let AA'B'C' be symmetric with respect to A A'B'X. Now show that S A'B'X and A'B'C' have 2XA'B' =<C'A'B', ZX=ZC', A'X= A'C', and are arranged in reverse order ($ 204), and hence that A ABC and A'B'C' have ZA=C'A'B', ZC=XC', and AC=A'C' (Ax. 5), and are similarly arranged. Then show that A ABC and A'B'C' are congruent, as proved above, and hence that AABC is symmetric with respect to A A'B'X (Ax. 5). Proposition 10. Mutually Equilateral Triangles 253. Theorem. If two triangles on the same sphere or on equal spheres are mutually equilateral, they are mutually equiangular and are either congruent or symmetric. Given ABC, A'B'C', two A on the same sphere or on equal spheres, with AB=A'B', BC=B'C', and CA=C'A'. Prove that ZA=ZA', ZB=ZB', ZC=ZC', and that A ABC and A'B'C' are either congruent or symmetric. Proof. Let O and O' be the centers of the spheres, and let a plane pass through each pair of vertices of each and the center of its sphere. Then in the trihedral 4 at 0 and O' the corresponding face { are respectively equal. $ 16, 10 Then the trihedral 40 and O' are equal (S 84), and hence the corresponding dihedral s are respectively equal ($ 81). .. ZA=ZA', ZB=ZB', ZC=ZC'. $ 190 Hence the A are either congruent or symmetric. $ 252 In the above figure the parts are arranged in the same order, so that the triangles are congruent. The parts might be arranged in reverse order, as in the SABC and A'B'X in the figure of $ 252, in which case the A ABC and A'B'C' would be symmetric. |