Mensuration of Solids. 3. If the diameters are 20 and 15, what will be the area included between the circumferences? Ans 137.445. 4. If the diameters are 16 and 10, what will be the area in cluded between the circumferences? Ans. 122.5224. 5 Two diameters are 21.75 and 9.5; required the area of the circular ring. Ans. 300.6609 6. If the two diameters are 4 and 6, what is the area of the Ans. 15.708 ring? The mensuration of solids is divided into two parts. 1st, The mensuration of the surfaces of solids: and We have already seen that the unit of measure for plane surfaces, is a square whose side is the unit of length (Bk. IV Def. 7). 2. A curve line which is expressed by numbers is also referred to an unit of length, and its numerical value is the number of times which the line contains the unit. If then, we suppose the linear unit to be reduced to a night line, and a square constructed on this line, this square will be the unit of measure for curved surfaces. 3. The unit of solidity is a cube, whose edge is the unit in which the lincar dimensions of the solid are expressed; and Mensuration of Solids. the face of this cube is the superficial unit in which the surface of the solid is estimated (Bk. VI. Th. xiii. Sch). 4 The following is a table of solid measure. Multiply the perimeter of the base by the altitude and the product will be the convex surface: and to this add the area of the bases, when the entire surface is required (Bk. VI. Th. i). EXAMPLES 1. Find the entire surface of the regular prism whose base is the regular polygon ABCDE and altitude AF, when each side of the base is 20 feet and the altitude AF, 50 feet. E AB+BC+CD+DE+EA=100; and AF-50: then (AB+BC+CD+DE+EA)×AF=convex surface Mensuration of Solids. 7. What is the solidity of a prism whose base is an equi lateral triangle, each side of which is 4 feet, the height of the prista being 10 feet? Ans. 69.282 solid ft. 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 PROBLEM III. 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 (Bk. VI. Th vi) EXAMPLES. 1. In the regular pentagonal pyramid S-ABCDE, the slant height SF is equal to 45, and each side of the base 18 15 feet required the convex surface, and also the entire surface. 15×5=75=perimeter of the base, 75 × 2211687.5 square feet area of convex surface. And 15-225: then 225 × 1.7204774-387.107415=the area area of the base 387.107415 Entire surface =2074.607415 square feet. Mensuration of Solids. 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. page 178. 152=225: and 225×1.7204774=387.107415= the area of the base: hence, 387.107415x20=7742.1483 solidity. 2. What is the solid contents of a cube whose side is 24 inches ? Ans. 13824 solid in. 3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? Ans. 21 solid ft. 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 ft. 6. What is the solidity of a square prism whose height is 5 feet, and each side of the base 1 foot? Ans 97 solid ft. Mensuration of Solids. 7. What is the solidity of a prism whose base is an equi lateral triangle, each side of which is 4 feet, the height of the pris being 10 feet? Ans. 69.282 solid ft. 8. What is the number of cubic or solid feet in a regular pentagonal prism of which the altitude is 15 feet and cach sile of the base 3.75 feet? Ans. 362.913 PROBLEM III. 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 (Bk. VI. Th vi) EXAMPLES. 1. In the regular pentagonal pyramid S-ABCDE, the slant height SF is equal to 45, and each side of the base 18 15 feet required the convex surface, and also the entire surface. 15×5 75 perimeter of the base, D 75 × 2212=1687.5 square feet area of convex surface. A And 152-225: then 225 × 1.7204774-387.107415=the area area of the base 387.107415 Entire surface =2074.607415 square feet. |