Proposition 6. Diagonal Plane of a Parallelepiped 103. Theorem. The plane passed through diagonally opposite edges of a parallelepiped divides the parallelepiped into two equivalent triangular prisms. Given the plane m passed through the opposite edges AA' and CC' of the parallelepiped AC'. Prove that AC' is divided into two equivalent triangular prisms ABC-B' and CDA-D'. Proof. Let WXYZ be a right section of AC'. $ 90 and faces AD' and BC' are congruent and II. $102 Now prism ABC-B' is equivalent to a rt. prism with base WXY and altitude AA', and prism CDA-D' is equivalent to a rt. prism with base YZW and altitude AA'. The rest of the proof is left for the student. $ 97 Exercises. Practical Measurements 1. Given that the three face angles at one of the vertices of a parallelepiped are 80°, 70°, and 75° respectively, find all the other angles in all the faces. 2. The three edges of the trihedral angle at one of the vertices of a rectangular parallelepiped are 10 in., 12 in., and 14 in. respectively. Find the area of the total surface of the parallelepiped. 3. The three face angles at one vertex of a parallelepiped are each 60°, and the three edges of the trihedral angle at that vertex are 6 in., 4 in., and 2 in. respectively. Find the area of the total surface to the nearest 0.01 sq. in. 4. In a rectangular parallelepiped the square of any diagonal is equivalent to the sum of the squares of any three edges which meet at one vertex. 5. In a box 6 in. deep and 12 in. wide, a wire 2 ft. long reaches from one corner to the diagonally opposite corner. Find the length of the box to the nearest 0.01 in. 6. The height of a rectangular parallelepiped is 22 in. and the length of the diagonal of the base is 30 in. Find the length of the diagonal of the parallelepiped. 7. The total area of the six faces of a cube is 108 sq. in. Find the length of the diagonal of the cube. 8. The diagonal of the face of a cube is √6. Find the diagonal of the cube. 9. The diagonal of a cube is 5 √3. Find the diagonal of a face of the cube. 10. A water tank is 4 ft. long, 3 ft. wide, and 2 ft. deep. How many square feet of zinc will be required to line the four sides and the base, allowing 2 sq. ft. for overlapping and for turning the top edge? 11. A square sheet of galvanized iron 8 ft. on a side leans against a wall and is inclined at an angle of 60° to the horizontal. What area of ground does the sheet protect from rain falling vertically? 12. Through a point P on the sheet of iron in Ex. 11, what line in the inclined plane will make the largest angle with the horizontal? Give the reason for your answer, and state the number of degrees in this angle. 13. A rectangular solid has for its lower base the ABCD and for its upper base the A'B'C'D', lettered in the corresponding way. What plane passing through the diagonal DB' is to AB, and what angle does the plane make with the lower base? 14. The length of the diagonal of a rectangular solid is 17 in. and the area of the total surface is 552 sq. in. Find the sum of the length, width, and height. 15. If the length, width, and height of a room are a, b, and c respectively, what is the total area of the four walls? of the walls, floor, and ceiling? 16. The outside measurements of a closed wooden box are 8 in., 10 in., and 12 in., and the area of the total inside surface is 376 sq. in. Find the thickness of the wood used in making the box. 17. The area of the total surface of a rectangular block is 1332 sq. in. and the length, width, and height are proportional to 4, 5, and 6. Find the edges. 18. The lower base of a cube which is √2 on an edge is ABCD, and the upper base is A'B'C'D', lettered in the corresponding way. If a plane passes through A', C', and B, what is the area of the AA'BC'? 19. The area of AA'BC' in Ex. 18 is what part of the area of the diagonal plane AB'C'D? 104. Unit of Volume. In measuring volumes, a cube whose edges are all equal to the unit of length is taken as the unit of volume. Thus, if we are measuring the contents of a box of which the dimensions are given in feet, we take 1 cu. ft. as the unit of volume. 105. Volume. The number of units of volume contained by a solid is called its volume. 106. Equivalent Solids. Two solids which have equal volumes are said to be equivalent. 107. Dimensions. The lengths of the three edges of a rectangular parallelepiped which meet at a common vertex are called its dimensions. 108. Volume of a Rectangular Parallelepiped. Assuming that the edges are commensurable (§ 100), suppose that 1, the length, contains 4 units; that w, the width, contains 2 units; and that h, the height, contains 5 units. Then V, the volume, contains 4 X 2 X 5, or 40, cubic units. In general, if there are l units of length, w units of width, and h units of height, then V = lwh; h that is, the volume of a rectangular parallelepiped is the product of its three dimensions. For on each square unit of base there is one cubic unit for every unit of height. Then, since there are lw square units of base (§ 25, 1), there are lw cubic units for every unit of height. Hence for h units of height, there are lwh cubic units of volume. The incommensurable case is considered in § 233. 109. Corollary. The volume of a rectangular parallelepiped is the product of its base and altitude. For, if B is the area of the base, B = lw, and hence V=Bh. Proposition 7. Volume of a Parallelepiped 110. Theorem. The volume of any parallelepiped is the product of the base and the altitude. F E H C R Given P, a parallelepiped with no two faces to each other; V, the volume; B, the area of the base; and h, the altitude. Prove that V=Bh. Proof. Produce the edge CD and the edges || to CD, and cut them perpendicularly by two || planes whose distance apart, EF, is equal to CD. We then have the oblique parallelepiped Q whose base is a . Produce FG and the edges | to FG, and cut them perpendicularly by two || planes whose distance apart, HI, is equal to FG. We then have the rectangular parallelepiped R. Since then PQ, and Q=R, P=R. $ 97 Ax. 5 Since then Also, P, Q, and R have the common altitude h. Let B' be the area of the base of Q, and B" that of R. B=B', and B' = B", $ 58 $ 26, 1 Ax. 5 Now the volume of R is B'h. $ 109 ..V=Bh. Ax. 5 |