Machinery's Handbook, 31st Edition
VOLUMES OF SOLIDS
93
Paraboloid Segment:
π 2 -- h R 2 r 2 + ( ) 1.5708 h R 2 r 2 + ( ) = =
V
π 8 -- h D 2 d 2 + ( ) 0.3927 h D 2 d 2 + ( ) = = Example: Find the volume of a segment of a paraboloid in which D = 5 in., d = 3 in., and h = 6 in. V 0.3927 h D 2 d 2 + ( ) 0.3927 6 × 5 2 3 2 + ( ) × = = 0.3927 6 × 34 × 80.11in 3 = =
R
r
D
d
h
Torus:
D V 2 π 2 Rr 2 19.739 Rr 2 = = π 2 4 --- Dd 2 2.4674 Dd 2 = = Area of surface A 4 π 2 Rr 39.478 Rr = = = π 2 Dd 9.8696 Dd = = Example: Find the volume and area of surface of a torus in which d = 1.5 in. and D = 5 in. V 2.4674 5 × 1.5 2 × 2.4674 5 × 2.25 × 27.76in 3 = = = A 9.8696 5 × 1.5 × 74.022in 2 = = Barrel: r R d = If the sides are bent to the arc of a parabola: V 0.209 h 2 D 2 Dd 3 ⁄ 4 d 2 + + ( ) = Example: Find the approximate contents of a barrel with sides bent to an arc of a circle, the inside dimensions of which are D = 60 cm, d = 50 cm, h = 120 cm. h D V = approximate volume If the sides are bent to the arc of a circle: V 1 12 --- π h 2 D 2 d 2 + ( ) = 0.262 h 2 D 2 d 2 + ( )
V 0.262 h 2 D 2 d 2 + ( ) 0.262 120 × 2 60 2 50 2 + ( × ) × = = 0.262 120 × 7200+2500 ( ) × 0.262 120 × 9700 × = = 304 968cm 3 , 0.305m 3 = =
Ratio of Volumes:
If d = base diameter and height of a cone, a paraboloid and a cyl inder, and the diameter of a sphere, then the proportions of the volumes of these solids are to each other as follows: Cone:Paraboloid :Sphere:Cylinder = ⅓ : ½ : ⅔ : 1
d
d
Example: Assume, as an example, that a cone, paraboloid, and cylinder each has a diameter of 2 inches, and that the height is 2 inches, and that the diameter of a sphere is 2 inches. Then the volumes, written in formula form, are as follows: Cone Paraboloid Sphere Cylinder 3.1416 2 2 × 2 × 12 ------------------ - : 3.1416 2 p ( ) 2 2 8 -------- -------- -- : 3.1416 2 3 6 ----------- : 3.1416 2 2 × 2 × 4 ------------------- = ⅓ : ½ : ⅔ : 1 × ---- × ----- × ----
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