MENSURATION OF SOLIDS. 1. THE mensuration of solids is divided into two parts: First. The mensuration of their surfaces; and, Second. The mensuration of their solidities. 2. We have already seen, that the unit of measure for plane surfaces is a square whose side is the unit of length. A curved line which is expressed by numbers is also referred to a unit of length, and its numerical value is the number of times which the line contains its unit. If then, we suppose the linear unit to be reduced to a right 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, the face of which is equal to the superficial unit in which the surface of the solid is estimated, and the edge is equal to the linear unit in which the linear dimensions of the solid are expressed (B. VII., P. 13, s. 1). The following is a table of solid measures: 1728 cubic inches = 1 cubic foot. 27 cubic feet 1 cubic yard. 4492; cubic feet = 1 cubic rod. OF POLYEDRONS, OR, SURFACES BOUNDED BY PLANES. 4. To find the surface of a right prism. Dsultiply the perimeter of the base by the altitude, and the pro duct will be the convex surface (B. VII., P. 1). To this add the area of the two bases, when the entire surface is required. Ex. 1. To find the surface of a cube, the length of each side being 20 feet. Ans. 2400 sq. ft. 2. To find the whole 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. 3. What must be paid for lining a rectangular cistern with leal, at 2d. a pound, the thickness of the lead being such as to require 7lbs. 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. 21. 3s. 10 d. 5. To find the surface of a right pyramid. Multiply the perimeter of the base by half the slant height, and the product will be the convex surface (B. VII., P. 4): to this add the area of the base, when the entire surface is required. Ec. 1. To find the convex surface of a right triangular pyramid, the slant height being 20 feet, and each side of the base 3 feet. Ans. 90 sq. ft. 2. What is the entire surface of a right pyramid, whose slant height is 15 feet, and the base a pentagon, of which each side is 25 feet ? Ans. 2012.798. 6. To find the convex surface of the frustum of a right pyramid. Multiply the half sum of the perimeters of the two bases by the slant height of the frustum, and the product will be the convex surface (B. VII., P. 4, c.) Er. 1. How many square feet are there in the convex surface of the frustum of a square pyramid, whose slant height is 10 feet, each side of the lower base 3 feet 4 inches, and each side of the upper base 2 feet 2 inches? Ans. 110 sq. ft. 2. What is the convex surface of the frustum of an heptagonal pyramid whose slant height is 55 feet, each side of the lower base 8 feet, and each side of the upper base 4 feet? Ans. 2310 sq. ft. 7. To find the solidity of a prism. 1. Find the area of the base. 2. Multiply the area of the base by the altitude, and the pro duct will be the solidity of the prism (B. VII., P. XIV). Ex. 1. What are the solid contents of a cube whose side is 24 inches ? Ans. 13824 2. 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. 213. 3. How many gallons of water, ale measure, will a cistern contain, whose dimensions are the same as in the last example ? Ans. 12947 4. Required the solidity of a triangular prism, whose height is 10 feet, and the three sides of its triangular base 3, 4, and 5 feet. Ans. 60. 8. To find the solidity of a pyramid. Multiply the area of the base by one-third of the altitude, and the product will be the solidity (B. VII., P. 17). Ex. 1. Required the solidity of a square pyramid, each side of its base being 30, and the altitude 25. Ans. 7500. 2. To find the solidity of a triangular pyramid, whose altitude is 30, and each side of the base 3 feet. Ans. 38.9711. 3. To find the solidity of a triangular pyramid, its altitude being 14 feet 6 inches, and the three sides of its base 5, 6, and 7 feet. Ans. 71.0352. 4. What is the solidity of a pentagonal pyramid, its altitude being 12 feet, and each side of its base 2 feet? Ans. 27.5276. 5. What is the solidity of an hexagonal pyramid, whose altitude is 6.4 feet, and each side of its base 6 inches? Ans. 1.38564. 9. To find the solidity of the frustum of a pyramid. Add together the areas of the two bases of the frustum, and a meun proportional between them, and then multiply the sum by one-third of the allilude (B. VII., P. 18). Ex. 1. To find the number of solid feet in a piece of timber, whose bases are squares, each side of the lower base being 15 inches, and each side of the upper base 6 inches, the altitude being 24 feet. Ans. 19.5. 2. Required the solidity of a pentagonal frustum, whose altitude is 5 feet, each side of the lower base 18 inches, and each side of the upper base 6 inches. Ans. 9.31925. DEFINITIONS. A B 10. A WEDGE is a solid bound G H ed by five planes : viz., a rectangle, ABCD, called the base of the wedge; two trapezoids A BHG, DCHG, which D are called the sides of the wedge, and which intersect each other in the edge GH; and the two triangles GDA, HCB, which are called the ends of the wedge. When AB, the length of the base, is equal to GH, the trapezoids A BIG, DCHIG, become parallelograms, and the wedge is then one-half the parallelopipedon described on the base ABCD, and having the same altitude with the wedge. The altitude of the wedge is the perpendicular let fall from any point of the line GH, on the base ABCD. 11. A RECTANGULAR PRISMOID is a solid resembling the frustum of a quadrangular pyramid. The upper and lower bases are rectangles, having their corresponding sides parallel, and the convex surface is made up of four trapezoids. The altitude of the prismoid is the perpendicular distance between its bases. TO FIND THE SOLIDITY OF THE WEDGE. Let L= AB, the length of G H the base, 1= GII, the length of the edge, b= BC, the breadth of the base, h = PG, the altitude of the wedge. N С P = AM. A M B Suppose AB, the length of the base, to be equal to GH, the length of the edge, the solidity will then be equal to half the parallelopipedon, same having the same base and the G H altitude (B. VII., P. 7). Hence, the solidity will be equal to blh (B. VII., P. 14). If the length of the base is N D greater than that of the edge, T let a section MNG be made A M B parallel to the end BCH. The wedge will then be divided into the triangular prism BCH-G, and the quadrangular pyramid G-AMIND. Then, the solidity of the prism }bhl; the solidity of the pyramid = {bh (L – 1); and their sum, juhl + Fbh(L-1)= 10231+ Zuh 2L - 10/21 = 16h(2L + 1). If the length of the base is less than the length of the edge, the solidity of the wedge will be equal to the difference between the prism and pyramid, and we shall have for the solidity of the wedge, i bhl — }bh(1 – L) = 16h31 – 16h21 + 7112L = 1bh(2L + 1). Ec. 1. If the base of a wedge is 40 by 20 feet, the edge 35 feet, and the altitude 10 feet, what is the solidity? Ans. 3833.33. 2. The base of a wedge being 18 feet by 9, the edge 20 feet, and the altitude 6 feet, what is the solidity ? Ans. 504. 12. To find the solidity of a rectangular prismoid. Add together the areas of the two bases and four times the area of a parallel section at equal distances from the bases : then multiply the sum by one-sixth of the altitude. For, let L and B denote the length 1' and breadth of the lower base, I and 3 b the length and breadth of the upper base, and m the length M and breadth of the section equidistant from the bases, and h the alti B tude of the prismoid. L Z |