Machinery's Handbook, 31st Edition
AREAAND VOLUME
71
Area and Volume The Prismoidal Formula.— A right prism is a three-dimensional figure composed of (two-dimensional) polygons that form the faces of the prism. The end faces are the bases, the other faces are the sides. The prismoidal formula is a general formula by which the volume of any prism, pyramid, or frustum of a pyramid may be found. A 1 , A 2 = end areas of the body A m = area of faces between the two end surfaces h = height of body Then, volume of the body is calculated as V 4 A m A 2 + + ( ) = . Pappus-Guldinus Rules.— A surface of revolution is generated when a curve is revolved about an external axis. (The curve must lie wholly on one side of the axis of revolution and in the same plane.) The mathematics for finding the curve’s length and the length of the path of the centroid involves calculus. Some surface areas and volumes of solids of revo- lution can then be determined by the rules of the Pappus-Guldinus theorems . The area of the resulting surface is equal to the product of the length of the generating curve and the distance traveled by the curve’s center of gravity, or centroid (see figure below). h 6 -- A 1
The volume of a solid body formed by the revolution of a surface FGHJ about axis KL equals the product of the surface area and the length of the path of its center of gravity about axis KL . C
A
5 ”
3 ”
B
D
Example: By means of the Pappus-Guldinus rules, the area and volume of a cylindrical ring, or torus, may be found. A torus is formed when a circle is rotated about an axis. The center of gravity of the circle is its center. Hence, with the dimensions given in the illustration, the length of the path of the center of gravity of the circle it travels is the circumference 2 π r = 2 × 3.1416 × 5 = 31.416 inches. Multiplying this path length by the circumference of the circle, which is 3.1416 × 3 = 9.4248 inches, gives 31.416 × 9.4248 = 296.089 square inches. This is the surface area of the torus. The volume of the torus equals the area of the circle, ------ d 2 4 = 0.7854 × 9 = 7.0686 square inches, multiplied by the path length of the center of gravity, which is 31.416, as before; hence, Volume 7.0686 31.416 × 222.067 cubic inches = =
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