185. Examples. 1. What is the entire surface of a cylinder whose altitude is 6 ft. and radius 2 ft.? Ans. 100.5312 sq. ft. 2. What is the entire surface of a cylinder whose altitude is 100 ft. and radius 20 ft.? Ans. 15079.68 sq. ft. 186. Problem. To find the area of the entire surface of a cone. Let r be the radius of the base of the cone, a the slant height, and k the area of the entire surface. 187. Examples. 1. What is the entire surface of a cone whose slant height is 10 ft. and radius 5 ft.? Ans. 235.62 sq. ft. 2. What is the entire surface of a cone whose altitude is 100 ft. and radius 25 ft.? Ans. 10059.1675 sq. ft. 188. Problem. To find the area of the entire surface of the frustum of a cone. Let be the radius of the lower base, r' be the S. N. 15. radius of the upper base, a the slant height, and k the area of the entire surface. πa (r+r') πr2 Tr'2 = the convex surface. the area of the lower base. the area of the upper base. •'. k = [a (r + r) + p2 + p2] • π of which the radius of the is 20 ft. 189. Examples. 1. Find the entire surface of the frustum of a cone radius of the lower base is 10 ft., the upper base is 6 ft., and slant height Ans. 1432.5696 sq. ft. 2. Find the entire surface of the frustum of a cone of which the radius of the lower base is 25 in., the radius of the upper base 12 in., and the slant height 36 in. Ans. 45.8368 sq. ft. (3) k 190. Problem. To find the area of the surface of a sphere. Let be the radius, d the diameter, c the circumference, and k the area. Then, by Geometry, (1) k 4 πρ. (2) k πd2. a 191. Examples. 1. The radius of a sphere is 10 ft.; required the Ans. 1256.64 sq. ft. area. 2. The diameter of a sphere is 25 ft.; required the Ans. 1963.5 sq. ft. area. 3. The circumference of a sphere is 100 in.; required Ans. 3183.0914 sq. in. the area. 4. The circumference of a sphere is ameter 20; required the area. 192. Problem. To find the area of a zone. By Geometry, the area of a zone is equal to the circumference of a great circle multiplied by the altitude of the zone. Let a denote the altitude of the zone, r the radius of the sphere, and k the area of the zone. ... k 2 πια. 62.832, and di Ans. 1256.64. 193. Examples. 1. What is the area of the torrid zone, calling its width 46° 56', and the earth a perfect sphere whose radius is 3956.5 mi.? Ans. 78333333. sq. mi. 2. What is the area of the two frigid zones if the polar circles are 23° 28′ from the poles? Ans. 16270370. sq. mi. 3. What is the area of the two temperate zones? Ans. 102109933. sq. mi. 194. Problem. To find the area of a spherical triangle. Let s = A + B + C, and 2 tri-rectangular triangle. Then, by Geometry, the .2 k = 1⁄2 πr2 (90°-2). B 8 In this formula, 2 is to be regarded as an 90° abstract number. Minutes and seconds are to be re duced to the decimal of a degree. 195. Examples. 1. Find the area of the spherical triangle whose angles are 60°, 80°, 100°, and the radius 3956.5 mi. Ans. 16392592 sq. mi. 2. Find the area of a spherical triangle whose sides are 70°, 90°, 100°, respectively, and radius 100 in. Ans. 10942.1928 sq. in. 196. Problem. To find the area of a spherical polygon. Let s be the sum of the angles, n the number of sides, k the area of the polygon, and r the radius of the sphere. Then, by Geometry, 197. Examples. 1. The sum of the angles of a spherical hexagon is 800°, the radius is 100 ft.; required the area. Ans. 13963. sq. ft. 2. Each angle of a spherical radius is 50 ft.; required the area. pentagon is 120°, the Ans. 2618. sq. ft. 3. The angles of a spherical polygon are 90°, 100°, 110°, 150°, respectively, the radius is 10 ft.; required the area. Ans. 157.08 sq. ft. 4. Each angle of a spherical decagon is 150°, the radius is 1 ft.; required the area. Ans. 1.0472 ft. k'e 2 198. Problem. To find the area of the surface of a regular polyhedron. Let e be one edge, n the number of faces, k' the area of a polygon whose side is 1, and similar to one face, and k the area of the entire surface. the area of one face. Article 168. .. k nk'e2. = 199. Examples. 1. What is the area of the entire surface of a tetrahedron whose edge is 10 ft.? Ans. 173.20508 sq. ft. entire surface of a hexaAns. 150 sq. ft. 2. What is the area of the hedron whose edge is 5 ft.? 3. What is the area of the hedron whose edge is 20 ft.? entire surface of an octaAns. 1385.64064 sq. ft. entire surface of a dodec Ans. 32.25895 sq. ft. 4. What is the area of the ahedron whose edge is 15 in.? 5. What is the area of the entire surface of an icosahedron whose edge is 100 in.? Ans. 601.4065 sq. ft. |