595. THEOREM. The volume of a rectangular parallelepiped is equal to the product of its base by its altitude. (See 594.) 596. COR. The volume of a cube is equal to the cube of its edge. 597. THEOREM. The volume of any parallelepiped is equal to the product of its base by its altitude. = Given: Parallelepiped R, whose base = B and alt. h. = On Proof: Prolong the edge AD and all edges | to AD. the prolongation of AD, take EF AD. Through E and F pass planes EG and FH, to EF, forming the right parallelepiped s. Again, prolong FI and all the edges to FI. On the prolongation of FI, take KL = FI. Through K and L pass planes KM and LN, to KL, forming the rectangular parallelepiped T. Consider EG the base of s, and EF its altitude, then R≈ S (?) (587). Also B B' (?) (374). Consider EP the base of S, and KM the base of T, and KL its altitude, then 8 T (?) (587). Also B'=c (?)(140). Hence, RT (Ax. 1); and B C (Ax. 1); and altitude of T h (?) (524). But volume of Tch (?) (595). = .. volume of R = B. h (Ax. 6). Q.E.D. 598. THEOREM. Two parallelepipeds having equal altitudes and equivalent bases are equivalent (Ax. 1). 601. THEOREM. Any two parallelepipeds are to each other as the products of their bases by their altitudes (?). 602. THEOREM. The volume of a triangular prism is equal to the product of its base by its altitude. Volume of ACD-F-B.h. Proof: Construct parallelepiped 48 having as three of its lateral edges AE, CF, DG. Vol. AS=ACRD·h (?)(597). Hence, volume of AS = But B ACRD h (Ax. 3). S volume of AS= volume of prism ACD-F (?) (588) and ACRDB (?) (132). .. volume of ACD-F= B·h (Ax. 6). Q. E.D. Ex. 1. Which rectangular parallelepiped contains the greater volume, one whose edges are 5, 7, 9, or one whose edges are 4, 6, 13? Ex. 2. The base of a prism is a right triangle whose legs are 8 and 12, and the altitude of the prism is 20. Find its volume. 603. THEOREM. The volume of any prism is equal to the product Vol. of prism AD=(R+S+T)h = B · h (Ax. 2). I Q.E.D. 604. THEOREM. Two prisms having equal altitudes and equivalent bases are equivalent. 605. THEOREM. Two prisms having equal altitudes are to each other as their bases. 606. THEOREM. Two prisms having equivalent bases are to each other as their altitudes. 607. THEOREM. Any two prisms are to each other as the products of their bases by their altitudes. ORIGINAL EXERCISES 1. How many faces has a parallelepiped? Edges? Vertices? How many faces has a hexagonal prism? Edges? Vertices? 2. Every lateral face of a prism is parallel to the lateral edges not in that face. 3. Every lateral edge of a prism is parallel to the faces that do not contain it. 4. Every plane containing one and only one lateral edge of a prism is parallel to all the other lateral edges. 5. Any lateral face of a prism is less than the sum of the other lateral faces. [Use fig. of 583.] 6. The diagonals of a rectangular parallelepiped are equal. Proof Pass the plane ACGE. This is a rectangle (?), etc. 7. The four diagonals of a parallelepiped bisect H each other. [First prove that one pair bisect each other, thus prove that any pair bisect each other, etc.] 8. Two triangular prisms are equal if their lateral faces are equal each to each. B 9. Any prism is equivalent to the parallelepiped having the same altitude and an equivalent base. 10. The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions. To Prove: AC2 = AE2 + ED2 + DC2. Proof: AD is the hypotenuse of rt. ▲ AED, and AC, of rt. A ACD. E 11. The diagonal of a cube is equal to the edge multiplied by √3. 12. The volume of a triangular prism is equal to half the product of the area of any lateral face by the perpendicular drawn to that face from any point in the opposite edge. [Use the fig. of 602.] 13. Every section of a prism made by a plane parallel to a lateral edge is a parallelogram. To Prove: LMRN a. Proof: LM is || to NR (?). LN and MR are each || to any edge. (Explain.) 14. Every polyhedron has an even number of face angles. B N The number or = the number of Proof: Consider the faces as separate polygons. sides of these polygons = double the number of edges of the polyhedron. (Explain.) But the number of sides of these polygons their angles, that is, the number of face angles... the number of face angles double the number of edges: = an even number (?). = 15. There is no polyhedron having fewer than 6 edges. 16. A room is 7 m. long, 5 m. wide, 3 m. high. Find its contents and its total area. 17. Find the volume, lateral area, and total area of an 8-in. cube. 18. A right prism whose height is triangle whose legs are 6 ft. and 8 ft. and total area of the prism. 12 ft. has for its base a right 19. Find the altitude of a rectangular parallelepiped whose base is 21 in. x 30 in., equivalent to a rectangular parallelepiped whose dimensions are 27 in. x 28 in. x 35 in. 20. A cube and a rectangular parallelepiped whose edges are 6, 16, and 18, have the same volumes. Find the edge of the cube. ห้ 21. Find the volume of a rectangular parallelepiped whose total area is 620 and whose base is 14 × 9. = 1808 cu. mito 22. How many bricks each 8 × 2 × 2 in. will be required to build a wall 22 × 3 × 2 ft. (not allowing for mortar)? 5 1 8 4 23. If a triangular prism is 20 in. high and each side of its base is 8 in., how many cubic inches does it contain? 320 24. Find the lateral area, total area, and volume of a regular hexagonal24 prism each side of whose base is 10 and whose altitude is 15.LA -900, TA = gen25. A box is 12 x 9 x 8 in. What is the length of its diagonal? 17 26. Each edge of a cube is 8 in. Find its diagonal. 27. The diagonal of a cube is 10 √3. Find its edge, volume, total area. 1000 600 28. A trench is 180 ft. long and 12 ft. deep, 7 ft. wide at the top and 4 ft. at the bottom. How many cubic yards of earth have been removed? 440u 29. A metallic tank, open at the top, is made of iron 2 in. thick; the internal dimensions of the tank are, 4 ft. 8 in. long, 3 ft. 6 in. wide, 4 ft. 4 in. deep. Find the weight of the tank if empty; if full of water. [Water weighs 621 lb. to the cu. ft. and iron is 7.2 times as heavy as water.] 30. The base of a right parallelepiped is a rhombus whose sides are each 25, and the shorter diagonal is 14. The height of the parallelepiped is 40. Find its volume and total surface. 31. If the diagonal of ́a cube is 12 ft., find its surface. 48x b 32. If the total surface of a cube is 1 sq. yd., find its volume in cu. ft. 33. A right prism whose altitude is 25 has for its base a triangle whose sides are 11, 13, 20. Find its lateral area, total area, and volume. 5422 4 2.11.3=66 you |