MENSURATION OF VOLUMES. 200. Problem. To find the volume of a prism. Let k be the area of the base, a the altitude, and v the volume. Then, by Geometry, v = ak. 201. Examples. 1. What is the volume of a regular hexagonal prism whose altitude is 20 ft., and each side of the base 10 ft.? Ans. 5196.1524 cu. ft. 2. What is the volume of a triangular prism whose altitude is 6 ft., and the sides of its base 3 ft., 4 ft., and 5 ft., respectively? Ans. 36 cu. ft. 3. What is the volume of a regular octagonal prism whose altitude is 120 ft., and each side of the base 20 ft.? Ans. 231764.5008 cu. ft. 202. Problem. To find the volume of a pyramid. Let be the area of the base, a the altitude, and v the volume. v = ak. 203. Examples. 1. What is the volume of a pyramid whose altitude is 15 ft., and whose base is a regular heptagon each side of which is 5 ft.? Ans. 454.23905 cu. ft. 2. What is the volume of a pyramid whose altitude is 21 in., and whose base is a triangle each side of which is 30 in.? Ans. 2727.98 cu. in. 204. Problem. To find the volume of the frustum of a pyramid. Let k and k, be the areas of the bases, a the altitude, and the volume. Then, by Geometry, = (1) v = } a (k + k1+V kk1). 1 If the bases are regular polygons whose sides are 8 and s', we shall have, by article 168, kk's2, and k1 k's'2, in which is given in the table of article 167, and (1) becomes. (2) va (82 + s′2 + 88′) k'. 205. Examples. 1. What is the volume of the frustum of a pyramid whose altitude is 9 ft., and whose bases are regular triangles, one side of the lower being 8 ft., and one side of upper, 5 ft.? Ans. 167.576 cu. ft. 2. What is the volume of the frustum of a pyramid whose altitude is 27 in., and the bases regular hexagons, the sides of which are 10 in. and 6 in., respectively? Ans. 4583.0064 cu. in. 206. Problem. To find the volume of a cylinder. Let r represent the radius, a the altitude, and v the volume. v = aπг2. 207. Examples. 1. What is the volume of a cylinder whose altitude is 50 in., and radius 15 in.? Ans. 20.453 cu. ft. 2. What is the volume of a cylinder whose altitude is 25 ft., and radius 4 ft.? Ans. 1256.64 cu. ft. 208. Próblem. To find the volume of a cone. Let be the radius of the base, a the altitude, and the volume. v = απ2. 209. Examples. 1. What is the volume of a cone whose altitude is 21 in., and radius 10 in.? Ans. 2199.12 cu. in. 2. What is the volume of a cone whose altitude is 30 ft., and radius is 10 ft.? Ans. 31416. cu. ft. 210. Problem. To find the volume of the frustum of a cone. Let and be the radii of the bases, a the altitude, and the volume. v = Fan (r2 + r22 + rr'). 211. Examples. 1. What is the volume of the frustum of a cone whose altitude is 15 ft., and the radii of whose bases are 9 ft. and 4 ft., respectively? Ans. 2089.164 cu.ft. 2. How many barrels will that cistern contain whose altitude is 8 ft., the diameter at the bottom 4 ft., and at the top 6 ft.? Ans. 37.8 bbl. 212. Formulas for the Sphere. Let be the radius, d the diameter, e the circumference, the area of the surface, and the volume 1. r = of a sphere, then, by Geometry, we have d = 2 r, C = πd, k 4 m2, rk. From which verify the following table of formulas: = d. 9. с с с 2 π = 2 r. с π 36 v k k 36 v = 2 πη. πd. = 11. sphere whose radius is 8 ft. 4 of the shell 3 ft. 6 in.? 12. C= 13. k 14. k 15. k 16. k 17. 18. 19. 20. v = v = v = v = = 213. Examples. 1. Calling the diameter of the earth 7913 mi., and the diameter of the sun 856,000, find the ratio of their surfaces, also the ratio of their volumes. 2. What is the volume of the shell of a hollow in., and the thickness Ans. 1951.1081 cu. ft. 214. Problem. To find the volume of a spherical sector. A spherical sector is the volume generated by the revolution of any circular sector, ABC, about any diameter, DE. By Geometry, the volume of a spherical sector is equal to the zone which forms its base, multiplied by one-third of the radius. Let a be the altitude of the zone, and r the radius. 2. The angle BCD, in the 20°, ACB is 35°, r = 20 ft.; = = 215. Examples. 1. The altitude of the zone which forms the base of a sector is 6 ft., the radius is 12 ft.; required the volume. Ans. 1809.5616 cu. ft. diagram of last article, is required the volume. Ans. 6134.25 cu. ft. v = the vol. generated by ABFG. A 216. Problem. To find the volume of a spherical segment. A spherical segment is the portion of a sphere in cluded between two parallel planes. r' Let BF perpendicular to DE, and " AG perpendicular to DE. r = the radius, d' CF, and d" CG. = the vol. generated by ABC=πr2α. B A D E B D F G C E |