BOOK XII. APPLICATIONS OF GEOMETRY TO THE MENSURATION OF SOLIDS. DEFINITIONS. 658. MENSURATION OF SOLIDS, or VOLUMES, is the process of determining their contents. The SUPERFICIAL CONTENTS of a body is its quantity of surface. The SOLID CONTENTS of a body is its measured magnitude, volume, or solidity. 659. The UNIT OF VOLUME, or SOLIDITY, is a cube, whose faces are each a superficial unit of the surface of the body, and whose edges are each a linear unit of its linear dimensions. 660. TABLE OF SOLID MEASURES. 1728 Cubic Inches make 1 Cubic Foot 661. To find the surface of a RIGHT PRISM. Multiply the perimeter of the base by the altitude, and the product will be the CONVEX surface (Prop. I. Bk. VIII.). To this add the areas of the two bases, and the result will be the ENTIRE surface. EXAMPLES. 1. Required the entire surface of a pentangular prism, having each side of its base, A B C D E, equal to 2 feet, and its altitude, A F, equal to 5 feet. 2 x 5 = 10; 10 X 550 square feet, [the surface required. 2. The altitude of a hexangular prism is 12 feet, two of its faces are each 2 feet wide, three are each 2 feet wide, and the remaining face is 9 inches wide; what is the convex surface of the prism? 3. Required the entire surface of a cube, the length of each edge being 25 feet. 4. Required, in square yards, the wall surface of a rectangular room, whose height is 20 feet, width 30 feet, and length 50 feet. Ans. 355 sq. yd. PROBLEM II. 662. To find the solidity of a PRISM. Multiply the area of its base by its altitude, and the product will be its solidity (Prop. XIII. Bk. VIII.). EXAMPLES. 1. Required the solidity of a pentangular prism, having each side of its base equal to 2 feet, and its altitude equal to 5 feet. 22 X 1.720486.88192; 6.88192 x 5 = 34.40960 cubic × [feet, the solidity required. 2. Required the solidity of a triangular prism, whose length is 10 feet, and the three sides of whose base are 3, 4, and 5 feet. Ans. 60. 3. A slab of marble is 8 feet long, 3 feet wide, and 6 inches thick; required its solidity. 4. There is a cistern in the form of a cube, whose edge is 10 feet; what is its capacity in liquid gallons? Ans. 7480.519 gallons. 5. Required the solid contents of a quadrilateral prism, the length being 19 feet, the sides of the base 43, 54, 62, and 38, and the diagonal between the first and second sides, 70 inches. Ans. 306.047 cu. ft. 6. How many cords in a range of wood cut 4 feet long, the range being 4 feet 6 inches high and 160 feet long? PROBLEM III. 663. To find the surface of a RIGHT PYRAMID. Multiply the perimeter of the base by half its slant height, and the product will be the CONVEX surface (Prop. XV. Bk. VIII.). To this add the area of the base, and the result will be the ENTIRE surface. 664. Scholium. The surface of an oblique pyramid is found by taking the sum of the areas of its several faces. EXAMPLES. 1. Required the convex surface of a pentangular pyramid, A BCDE-S, each side of whose base, A B C D E, is 5 feet, and whose slant height, SM, is 20 feet. 5 X 525; 25 X 22 250 square M = [feet, the surface required. S E A 2. What is the entire surface of a triangular pyramid, of which the slant height is 18 feet, and each side of the base 42 inches? Ans. 99.804 sq. ft. 3. Required the convex surface of a triangular pyramid, the slant height being 20 feet, and each side of the base. 3 feet. 4. What is the entire surface of a quadrangular pyramid, the sides of the base being 40 and 30 inches, and the slant height upon the greater side 20.04, and upon the less side 20.07 feet? Ans. 125.308 ft. PROBLEM IV. 665. To find the surface of a FRUSTUM OF A RIGHT PYRAMID. Multiply half the sum of the perimeters of its two bases by its slant height, and the product will be the CONVEX surface (Prop. XVII. Bk. VIII.); to this add the areas of the two bases, and the result will be the ENTIRE surface. EXAMPLES. 1. What is the entire surface of a rectangular frustum whose slant height is 12 feet, and the sides of whose bases are 5 and 2 feet? 5 x 4 = 20; 2 X 4 = 8; 20+8=28; 28 x 12=168; 522229; 168+29 197 sq. ft., area required. 2. Required the convex surface of a regular hexangular frustum, whose slant height is 16 feet, and the sides of whose bases are 2 feet 8 inches and 3 feet 4 inches. 3. What is the entire surface of a regular pentangular frustum, whose slant height is 11 feet, and the sides of whose bases are 18 and 34 inches? PROBLEM V. Ans. 136.849 sq. ft. 666. To find the solidity of a PYRAMID. Multiply the area of its base by one third of its altitude (Prop. XX. Bk. VIII.). 285 2. What is the solidity of a hexangular pyramid, the altitude of which is 9 feet, and each side of the base 29 inches? 3. What is the solidity of a square pyramid, each side of whose base is 30 feet, and whose perpendicular height is 25 feet? Ans. 7500. 4. Required the solid contents of a triangular pyramid, the perpendicular height of which is 24 feet, and the sides of the base 34, 42, and 50 inches. Ans. 39.2354 cu. ft. PROBLEM VI. 667. To find the solidity of a FRUSTUM OF A PYRAMID. Add together the areas of the two bases and a mean proportional between them, and multiply that sum by one third of the altitude of the frustum (Prop. XXI. Bk. VIII.). EXAMPLES. 1. Required the solidity of the frustum of a quadrangular pyramid, the sides of whose bases are 3 feet and 2 feet, and whose altitude is 15 feet. 3×3=9;2×2=4; √9×4=6 (Prop. IV. Bk. II.); (9+4+6) × fé = 95 cu. ft., solidity required. 2. How many cubic feet in a stick of timber in the form of a quadrangular frustum, the sides of whose bases are 15 inches and 6 inches, and whose altitude is 20 feet? 3. Required the solid contents of a pentangular frustum, whose altitude is 5 feet, each side of whose lower base is 18 inches, and each side of whose upper base is 6 inches. Ans. 9.319 cu. ft. 4. Required the solidity of the frustum of a triangular pyramid, the altitude of which is 14 feet, the sides of the lower base 21, 15, and 12, and those of the upper base 14, 10, and 8 feet. Ans. 868.752 cu. ft. |