Machinery's Handbook, 31st Edition
88
VOLUMES OF SOLIDS Volumes of Solids
Cube:
Diagonal of cube face d s 2 = = Diagonal of cube D 3 d 2 2
d
----- s 3 1.732 s = = = =
D
s s
s
Volume V s 3 = = s V 3 = Side Example: The side of a cube s measures 9.5 centimeters. Find its volume. Volume V s 3 9.5 3 9.5 9.5 × 9.5 × 857.375cm 3 = = = = = Example: The volume of cube is 231 cubic centimeters. What is the length of the side? s V 3 231 3 6.136cm = = = Rectangular Prism:
Volume
V abc = =
b
V bc = ---
V ac = ---
V ab = ---
a
b
c
c
a
Example: In a rectangular prism, a = 6 in., b = 5 in., c = 4 in. Find the volume. V abc 6 5 × 4 × 120in 3 = = = Example: What should the height of a box be if it is to contain 25 cubic feet and if it is 4 feet long and 2 1 ⁄ 2 feet wide? Here, a = 4, c = 2.5, and V = 25. Then, b height V --- 2.5 ft = = = = =
25 4 2.5 × -------- 25 10
ac ---
General Right Prism:
A h = edge length A = area of end surface V = Ah The area A of the end surface is found by the formulas for areas of plane figures on the preceding pages. Example: A right prism having for its base a regular hexagon with a side s of 7.5 centimeters is 25 centimeters high. Find the volume. Area of hexagon A 2.598 s 2 2.598 56.25 × 146.14 cm 2 = = = = Volume of prism Ah 25 146.14 × 3653.5 cm 3 = = = Right Pyramid: h
1 3 ( )
Volume V = =
(Base area × h )
h A square pyramid is pictured. In general, any right pyramid of height h , whose base is a regular, n -sided polygon of side length s has volume: V nsrh 6 ------ nsh 6 ----- R 2 s 2 4 – -- = = where r is the radius of the circle inscribed in the base, and R is the radius of the circle circumscribed on the base. Example: A pyramid having a height of 5 feet has a base formed by a square, the sides of which are 3 feet. Find the volume. Area of base 3 × square feet, = = h = 5 feet Volume V = = 3 9 ( 1 ⁄ 3 )(Base area × h) = 1 ⁄ 3 × 9 × 5 A
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online