Machinery's Handbook, 31st Edition
VOLUMES OF SOLIDS
91
Sphere:
4 π r 3 3 ------ 4 π r 2 3 V 4 π ---- 3
π
d 3
6 ----- 4.1888 r 3
0.5236 d 3
Volume
V = =
= =
=
r
π d 2 = =
12.5664 r 2
3.1416 d 2
A = =
=
Surface area
0.6204 V 3
=
=
r
d
Example: Find the volume and the surface area of a sphere 6.5 centimeters diameter. V 0.5236 d 3 0.5236 6.5 3 × 0.5236 6.5 × 6.5 6.5 × × 143.79 cm 3 = = = = A 3.1416 d 2 3.1416 6.5 2 × 3.1416 6.5 × 6.5 × 132.73 cm 2 = = = = Example: The volume of a sphere is 64 cubic centimeters. Find its radius. r 0.6204 64 3 0.6204 4 × 2.4816cm = = = Spherical Sector: h c r V 2 π r 2 h 3 -------- 2.0944 r 2 h = = A 3.1416 r 2 h ⁄ 2 c + ( ) = total area of conical and spherical surface = c 2 h 2 r h – ( ) = Example: Find the volume of a sector of a sphere with a 6-inch diameter ( r = 3 inches) and 1.5-inch height h . Also find the length of chord c . V 2.0944 r 2 h 2.0944 3 2 × 1.5 × 2.0944 9 × 1.5 × 28.27 in 3 = = = = c 2 h 2 r h – ( ) 2 1.5 2 3 × – 1.5 ( ) 2 6.75 2 2.598 × 5.196 in = = = = = Spherical Segment: h c r V = volume A = area of spherical surface V 3.1416 h 2 r h 3 – -- 3.1416 h c 2 8 --- h 2 6 + --- = = c 2 4 h 2 + 8 h = ---------- Example: A segment of a sphere has the following dimensions: h = 50 millimeters; c = 125 millimeters. Find the volume V and the radius of the sphere of which the segment is a part. V 3.1416 50 × 125 2 8 ------ 50 2 6 + ---- × 157.08 15,625 8 --------- 2500 6 + ------ × 372,247 mm 3 372 cm 3 = = = = A 2 π rh 6.2832 rh 3.1416 c 2 4 --- h 2 + = = = c 2 h 2 r h – ( ) = r
125 2 4 50 2 × + 8 50 × ------------------
15,625 10,000 + 400 --------------------
25,625 400 -------- 64 mm =
=
=
=
r
Ellipsoid:
V 4 π 3
--- abc 4.1888 abc = = In an ellipsoid of revolution, or spheroid, where c = b : V 4.1888 ab 2 = Example: Find the volume of a spheroid in which a = 5 in., and b = c = 1.5 in. V 4.1888 5 × 1.5 2 × 47.124in 3 = =
b
c
a
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online