Machinery's Handbook, 31st Edition
VOLUMES OF SOLIDS
89
Frustum of Pyramid:
A 1
-- A 1 A 2 A 1 A 2 + + = = ( )
Volume V h 3
h
A 2
Example: The pyramid in the previous example is cut off 4.5 feet from the base, and the upper part removed. The sides of the rectangle forming the top surface of the frustum are, then, 1 and 1.5 feet long, respectively. Find the volume of the frustum. Area of top A 1 1 1.5 × ft 2 = = = Area of base A 2 2 3 × 6 ft 2 = = = V 4 5 3 ----- 1.5 6 1.5 6 × + + ( ) 1.5 7.5+ 9 ( ) 1.5 10.5 × 15.75 ft 3 = = = = × 1.5 Wedge:
V 2 a c + ( ) bh 6 -------------- = =
Volume
c
A wedge has five faces, two of them triangles and three trapezoids. Example: Let a = 4 inches, b = 3 inches, and c = 5 inches. The height h = 4.5 inches. Find the volume in in 3 and cm 3 . V 2 a c + ( ) bh 6 -------------- 2 4+5 ( × ) 3 4.5 × × 6 ------------------------- 175.5 6 ------- 29.25 in 3 = = = =
h
cm 3 in 3 -----
b
a
×
29.25 in 3 16.387064
479.32162 cm 3
=
=
V
Cylinder:
d Volume V 3.1416 r 2 h 0.7854 d 2 h = = = Surface area of open-ended cylinder S 6.2832 rh 3.1416 dh = = = Total surface area A of closed cylinder: A 6.2832 r r h + ( ) 3.1416 d 1 ⁄ 2 d h + ( ) = = Example: Diameter d of a cylinder is 2.5 inches. Height h is 20 inches. Find the volume and the area of the open-ended cylindrical surface S . V 0.7854 d 2 h 0.7854 2.5 2 × 20 × 0.7854 6.25 × 20 × 98.17in 3 = = = = S 3.1416 dh 3.1416 2.5 × 20 × 157.08in 2 = = = Cylinder Portion: r h
Volume V 1.5708 r 2 h = =
1 h 2 + ( ) 1 h 2 + ( )
=
0.3927 d 2 h
h 2
h 1
S 3.1416 r h 1 h 2 + ( ) = = 1.5708 d h 1 h 2 + ( ) =
Surface area
r
d
Example: A cylinder 125 millimeters in diameter is cut off at an angle, as shown in the illustration. Dimension h 1 = 100 mm, and h 2 = 150 mm. Find the volume and the surface area S of the cylinder. V 0.3927 d 2 h 1 h 2 + ( ) 0.3927 125 2 × 100+ 150 ( ) × = = 0.3927 15,625 × 250 × 1 533 984 mm 3 , , 1534 cm 3 = = = S 1.5708 d h 1 h 2 + ( ) 1.5708 125 × 250 × = = 49 087.5 mm 2 , 490.9 cm 2 = =
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online