3. Given the circumference of a circle = 150 rds.; required r, d, and k. 1000 sq. rds.; re 4. Given the area of a circle quired r, d, and c. 5. Find the diameter of a circle whose area is equal to that of a regular decagon, each side of which is 10 ft. Ans. 31.3. 6. The radius of a circle is 10 ft., the diagonals of an equal parallelogram are 24 ft. and 30 ft.; required their included angle. Ans. 60° 46′ 17′′. 7. The radii of two concentric circles are r and r'; find the area of the ring included by their circumferences. Ans. (r) (r — r'). 172. Problem. To find the area of a sector of a circle. Let a be the arc of a sector, d the degrees in the arc, r the radius, and k the area. (2) a = ra. dar 180 (3) k 173. Examples. 1. Find the area of a sector whose arc is 40° and radius is 10 ft. Ans. 34.907 sq. ft. 2. Find the area of a sector whose arc is 60° 24′ 30′′ and radius is 100 rds. Ans. 5271.64 sq. rds. 3. The area of a sector is 345 sq. ft., the radius is 20 ft.; required the arc. Ans. 98° 50′ 06′′. 4. The area of a sector is 1000 sq. rds., the arc is 30° 45'; required the radius. Ans. 61.04 rds. 174. Problem. To find the area of a segment of a circle. Let d be the degrees in the arc of the segment, r the radius, and k the area. ત If d is greater than 180, sin d is negative, and the second term in the value of k becomes positive, as it should, since, in this case, the segment is equal to the corresponding sector plus the triangle. 175. Examples. 1. Find the area of the segment of a circle whose arc is 36° and radius 10 ft. Ans. 2.027 sq. ft. 2. Find the area of a segment whose chord is 36 ft. and radius 30 ft. Ans. 147.30 sq. ft. 3. Find the area of a segment whose altitude is 36 rds, and radius 50 rds. Ans. 2545.85 sq. rds. 4. The arc of a segment is 147°28'46", the altitude is 36 rds.; required the area. Ans. 2545.85 sq. rds. 176. Problem. To find the area of an ellipse. Let a be the semi-major axis, and b the semi-minor axis. Then, Ray's Analytic Geometry, article 446, πab. 2 k's2 177. Examples. 1. The semi-axes of an ellipse are 10 in. and 7 in.; required the area. Ans. 219.912 sq. in. 2. The area of an ellipse is 125 sq. rds.; find the axes if they are to each other as 3 is to 2. Ans. 15.45; 10.30. k 178. Problem. To find the area of the entire surface of a right prism. Let p be the perimeter of the base, a the altitude, s one side of the base, k' the area of a polygon similar to the base, each side of which is unity, article 167, and k the area of the entire surface. = k the areas of the bases. Article 168. ap + 2 k's2. u 179. Examples. 1. What is the entire surface of a right prism whose altitude is 20 ft., and base a regular octagon each side of which is 10 ft.? Ans. 2565.68542 sq. ft. 2. What is the entire surface of a right hexagonal prism whose altitude is 12 ft., and each side of the base is 6 ft.? Ans. 619.0614864 sq. ft. 3. What is the entire surface of a right prism whose altitude is 15 in., and base a regular triangle each side of which is 3 in.? Ans. 142.7942286 sq. in. ξαρ 180. Problem. To find the area of the surface of a regular pyramid. Let p be the perimeter of the base, a the slant height, s one side of the base, k' and k as in the last problem. the convex surface. the area of the base. 181. Examples. 1. What is the entire surface of a regular pyramid whose slant height is 12 ft., and base a regular triangle each side of which is 5 ft.? Ans. 100.82532 sq. ft. 2. What is the entire surface of a right pyramid whose slant height is 100 ft., and base a regular decagon each side of which is 20 ft.? Ans. 13077.68352 sq. ft. 182. Problem. To find the entire surface of a frustum of a right pyramid. Let p be the perimeter of the lower base, p' the perimeter of the upper base, a the slant height, s one side of the lower base, s' one side of the upper base, k' and k as in Art. 178. the convex surface. a(p+p') the area of lower base. ... k = a (p + p') + k' (82 + 8′2). = 183. Examples. 1. What is the entire surface of a frustum of a pyramid whose slant height is 12 ft., and the bases regular decagons whose sides are 8 ft. and 5 ft., respectively? Ans. 1464.78458 sq. ft. 2 πγα = 2 πr2 2. What is the entire surface of a frustum of a pyramid whose slant height is 15 ft., and the bases regular hexagons whose sides are 10 ft. and 6 ft., respectively? Ans. 1073.338 sq. ft. 184. Problem. To find the area of the entire surface of a cylinder. Let be the radius of the cylinder, a its altitude, and k the area of the entire surface. the convex surface. the area of the bases. ... k2 πr (a + r). |