MENSURATION OF BROKEN AND CURVED SUR- To find the area of the entire surface of a right prism. 109. From the principle demonstrated in Book VII., Prop. I., we may write the following RULE.-Multiply the perimeter of the base by the altitude, the product will be the area of the convex surface; to this add the areas of the two bases; the result will be the area required. Examples. 1. Find the surface of a cube, the length of each side being 20 feet. Ans. 2400 sq. ft. 2. 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 sq. ft. To find the area of the entire surface of a right pyramid. 110. From the principle demonstrated in Book VII., Prop. IV., we may write the following RULE.-Multiply the perimeter of the base by half the slant height; the product will be the area of the convex surface; to this add the area of the base; the result will be the area required. Examples. 1. 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 whose slant height is 27 feet, and the of which each side is 25 feet? a right pyramid, base a pentagon Ans. 2762.798 sq. ft. To find the area of the convex surface of a frustum of a right pyramid. 111. From the principle demonstrated in Book VII., Prop. IV., S., we may write the following RULE.-Multiply the half sum of the perimeters of the two bases by the slant height; the product will be the area required. Examples. 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 a 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. 112. Since a cylinder may be regarded as a prism whose base has an infinite number of sides, and a cone as a pyramid whose base has an infinite number of sides, the rules just given may be applied to find the areas of the surfaces of right cylinders, cones, and frustums of cones, by simply changing the term perimeter to circumfer ence. Examples. 1. What is the convex surface of a cylinder, the diameter of whose base is 20, and whose altitude 50? Ans. 3141.6. 2. What is the entire surface of a cylinder, the altitude being 20, and diameter of the base 2 feet? Ans. 131.9472 sq. ft. 3. Required the convex surface of a cone, whose slant height is 50 feet, and the diameter of its base 8 feet. Ans. 667.59 sq. ft. 4. Required the entire surface of a cone, whose slant height is 36, and the diameter of its base 18 feet. Ans. 1272.348 sq. ft. 5. Find the convex surface of the frustum of a cone, the slant height of the frustum being 12 feet, and the circumferences of the bases 8.4 feet and 6 feet. Ans. 90 sq. ft. 6. Find the entire surface of the frustum of a cone, the slant height being 16 feet, and the radii of the bases 3 feet and 2 feet. Ans. 292.1688 sq. ft. To find the area of the surface of a sphere. 113. From the principle demonstrated in Book VIII., Prop. X., C. 1, we may write the following RULE. Find the area of one of its great circles, and multiply it by 4; the product will be the area required. Examples. 1. What is the area of the surface of a sphere, whose radius is 16? Ans. 3216.9984. 2. What is the area of the surface of a sphere, whose radius is 27.25 ? Ans. 9331.3374. To find the area of a zone. 114. From the principle demonstrated in Book VIII., Prop. X., C. 2, we may write the following RULE.-Find the circumference of a great circle of the sphere, and multiply it by the altitude of the zone; the product will be the area required. Examples. 1. The diameter of a sphere being 42 inches, what is the area of the surface of a zone whose altitude is 9 inches? Ans. 1187.5248 sq. in. 2. If the diameter of a sphere is 12 feet, what will be the surface of a zone whose altitude is 2 feet? Ans. 78.54 sq. ft. To find the area of a spherical polygon. 115. From the principle demonstrated in Book IX., Prop. XIX., we may write the following RULE. From the sum of the angles of the polygon, subtract 180° taken as many times, less two, as the polygon has sides, and divide the remainder by 90°; the quotient will be the spherical excess. Find the area of a great circle of the sphere, and divide it by 2; the quotient will be the area of a tri-rectangular triangle. Multiply the area of the tri-rectangular triangle by the spherical excess, and the product will be the area required. This rule applies to the spherical triangle, as well as to any other spherical polygon. Examples. 1. Required the area of a triangle, described on a sphere whose diameter is 30 feet, the angles being 140°, 92°, and 68°. Ans. 471.24 sq. ft. 2. What is the area of a polygon of seven sides, described on a sphere whose diameter is 17 feet, the sum of the angles being 1080°? Ans. 226.98. 3. What is the area of a regular polygon of eight sides, described on a sphere whose diameter is 30 yards, each angle of the polygon being 140°? Ans. 157.08 sq. yds. MENSURATION OF VOLUMES. To find the volume of a prism. 116. From the principle demonstrated in Book VII., Prop. XIV., we may write the following RULE.-Multiply the area of the base by the altitude; the product will be the volume required. Examples. 1. What is the volume of a cube, whose side is 24 inches? Ans. 13824 cu. in. 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. 21 cu. ft. |