Find the area of the base, and multiply that area by the perpendicular height, and take ⅓ of the product.
Example.—Required the solidity of a square pyramid, each side of its base being 30, and its perpendicular height 25.
30 × 30 = 900, area of base.
900 × 25 3 = 7500, solidity required.
To find the solidity of the frustrum of a cone, or pyramid.
Add into one sum the areas of the two ends, and the mean proportional between them: take ⅓ of that sum for the mean area, which multiply by the perpendicular height, or length of the frustrum.
Note.—To find a mean proportional.
As one of the sides of the base is to the homologous, or corresponding side of the other end, so is the area of the base to the mean proportional required.
Example.—Required the number of solid feet in a piece of timber, whose bases are squares, each side of the greater end being 15 inches, and each side of the less end 6 inches; also the length of the perpendicular altitude 24 feet.
15 × 15 = 225, area of the base.
6 × 6 = 36, area of the top.
As 15: 6 :: 225: 90, mean proportional.
24 feet = 288 inches.
(225 + 36 + 90) × 288 3 = 33696 cubic inches = 19½ cubic feet.
To find the surface of a sphere, or any segment.