Machinery's Handbook, 31st Edition
90
VOLUMES OF SOLIDS
Segment of a Cylinder:
b In the formulas below, segment area A and arc length l refer to the shaded segment of the circle. The formulas of these dimensions, found on page 75, assume the central angle measure is known. Volume V 2 ⁄ 3 a 3 b segment area ( ± × ) L = = Surface area S ad b arc length area ( ± × ) L = = Use + when shaded segment is greater than or equal to one-half the base circle area, use – when it is less than the base circle area. r b ± ------ r b ± ------
a
r
d
d
Example: Find the volume of a cylinder segment of a 2-inch high cylinder with a diameter of 5 inches when it is cut so that line AC passes through the center of the base circle—that is, the base area is a half-circle. In this case, b = 0, and a = r ; that is, a = 2.5 inches. Thus, ABC is a semicircle, and area ABC = 3.1416 × r 2 × 1 ⁄ 2 = 3.1416 × (2.5) 2 × 1 ⁄ 2 = 9.82 in 2 . V 2 -- 15.625 × 0.8 × 8.33in 3 = = =
2 3 -- 2.5 3 × 0 9.82 × +
2.5+0 -------- 2 3
Hollow Cylinder:
3.1416 h R 2 r 2 – ( ) 3.1416 ht 2 R t – ( )
0.7854 h D 2 d 2 – ( )
Volume
V = =
=
D
3.1416 ht D t – ( ) = 3.1416 ht d t + ( ) =
= =
3.1416 ht 2 r t + ( ) 3.1416 ht R r + ( )
t
R r
1.5708 ht D d + ( ) = = Example: A cylindrical shell, 28 cm high, is 36 cm in outside diameter, and 4 cm thick. Find its volume. V 3.1416 ht D t – ( ) 3.1416 28 × 4 36– 4 ( ) × 3.1416 28 × 4 × 32 × = = = 11 259.5cm 3 , =
h
d
Cone:
V 3.1416 r 3 ------------- 1.0472 r 2 h 0.2618 d 2 h = = = = 2 h
Conical surface area A 3.1416 r r 2 h 2 + 3.1416 rs = = = 1.5708 ds = s r 2 h 2 + d 2 4 --- h 2 + = = Volume Example: Find the volume and surface area of a cone, the base of which is a circle of 6 inches diameter and the height of which is 4 inches. V 0.2618 d 2 h 0.2618 6 2 × 4 × 0.2618 36 × 4 × 37.7 in 3 = = = = A 3.1416 r r 2 h 2 + 3.1416 3 × 3 2 4 2 + × 9.4248 25 × = = = 47.124 in 2 = Frustum of Cone: V = volume A = area of conical surface V 1.0472 h R 2 Rr r 2 + + ( ) 0.2618 h D 2 Dd d 2 + + ( ) = = A 3.1416 s R r + ( ) 1.5708 s D d + ( ) = = a R r = – s a 2 h 2 + R r – ( ) 2 h 2 + = =
Example: Find the volume of a frustum of a cone of the following dimensions: D = 8 centimeters; d = 4 centimeters; h = 5 centimeters. V 0.2618 5 8 2 8 4 4 2 + × + ( ) × 0.2618 5 64 32 16 + + ( ) × = = 0.2618 5 × 112 × 146.61cm 3 = =
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