Mensuration of Solids. AB2x tabular number-area ABCDE, (see page 136), that is, 20 x tabular number, or 400 x 1,720477= 6SS,1908 the area ABCDE. Then, convex surface=5000 lower base upper base entire surface square feet. 2. What is the surface of a cube, the length of each side being 20 feet? Ans. 2400 sq. ft. 3. Find the entire surface of a triangular prism, whose base is an equilateral triangle, having each of its sides equal to 18 inches, and altitude 20 feet. Ans. 91,949 sq. ft. 4. What is the convex surface of a regular octagonal prism, the side of whose base is 15 and altitude 12 feet? Ans. 1440 sq. ft. 5. What must be paid for lining a rectangular cistern with lead at 2d a pound, the thickness of the lead being such as to require 77b. for each square foot of surface: the inner dimensions of the cistern being as follows: viz. the length 3 feet 2 inches, the breadth 2 feet 8 inches, and the depth 2 feet 6 inches? Ans. £ 2 3s. 10 d. Mensuration of Solids. PROBLEM II. 6. To find the solidity of a prism. RULE. Multiply the area of the base by the perpendicular height, and the product will be the area. EXAMPLES. 1. What is the solidity of a regular pentagonal prism whose altitude is 20, and each side of the base 15 feet. To find the area of the base we have by Problem VIII of ; I 15225: and 225 x 1,7204774-387,107415= the area of the base: hence, 387,107415×20=7742,1483=solidity. 2. What is the solid content of a cube whose side is 24 inches? Ans. 13824 solid inches. 3. How many cubic feet in a block of marble, of QUEST.-6. How do you find the solidity of a prism? Mensuration of Solids. which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? Ans. 21 solid feet. 4. How many gallons of water, ale measure, will a cistern contain whose dimensions are the same as in the last example? Ans. 12917. 5. Required the solidity of a triangular prism whose altitude is 10 feet, and the three sides of its triangular base, 3, 4, and 5 feet? Ans. 60 solid feet. 6. What is the solidity of a square prism whose height is 5 feet, and each side of the base 14 foot? Ans. 9 solid feet. 7. What is the solidity of a prism, whose base is an equilateral triangle, each side of which is 4 feet, the height of the prism being 10 feet? Ans. 69,282 solid feet. 8. What is the number of cubic or solid feet in a regular pentagonal prism of which the altitude is 15 feet and each side of the base 3,75 feet? Ans. 362,913. Mensuration of Solids. PROBLEM III. 7. To find the surface of a regular pyramid. RULE. Multiply the perimeter of the base by half the slant height, and the product will be the convex surface: to this add the area of the base, if the entire surface is required. 1. In the regular pentagonal pyramid S-ABCDE, the slant height SF is equal to 45, and each side of the base is 15 feet: required the convex surface, and also the entire surface. 15×5 75 perimeter of the base 75 × 221=1687,5 square feet= area of convex surface. S E B A 2. What is the convex surface of a regular triangular QUEST.-7. How do you find the surface of a regular pyramid? Mensuration of Solids. pyramid the slant height being 20 feet and each side of the base 3 feet. Ans. 90 sq. ft. 3. What is the entire surface of a regular pyramid whose slant height is 15 feet, and the base a regular pentagon, of which each side is 25 feet. Ans. 2012,798 sq. ft. PROBLEM IV. 8. To find the convex surface of the frustum of a regular pyramid. RULE. Multiply half the sum of the perimeters of the two bases by the slant height of the frustum, and the product will be the convex surface. EXAMPLES. 1. In the frustum of the regular pentagonal pyramid each side of the lower base is 30 and each side of the upper base is 20 feet, and the slant height fF is equal to 15 feet. What is the convex surface of the frustum? Ans. 1875 sq. ft. F QUEST.-8. How do you find the convex surface of the frustum of a regular pyramid ? |