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17.4 Volumes of Solids: Cubic Measure
A cubic unit is a cube whose edge is 1 unit long. Thus, a cubic inch is a cube whose side is 1 in long (Fig. 17-21).
CHAPTER 17 Extending Plane Geometry into Solid Geometry
4 1 in 1 in 1 in One Cubic Inch 5
Fig. 17-21
Fig. 17-22
The volume of a solid is the number of cubic units that it contains. Thus, a box 5 units long, 3 units wide, and 4 units high has a volume of 60 cubic units; that is, it has a capacity or space large enough to contain 60 cubes, 1 unit on a side. (See Fig. 17-22.) Here are some formulas for the volumes of solids. In these formulas, V is the volume of the solid, B is the area of a base, and h is the distance between the bases or between the vertex and a base. In volume formulas, the volume is in cubic units, the unit being the same as that used for the dimensions. Thus, if the edge of a cube measures 3 meters, its volume is 27 cubic meters. 1. Rectangular solid (Fig. 17-23): V lwh 2. Cylinder (Fig. 17-24): V Bh or V pr2h
h w l
h r r
Fig. 17-23
Fig. 17-24
3. Prism (Fig. 17-25): V 4. Cube (Fig. 17-26): V
Bh e3
e h B B h e e
Fig. 17-25
Fig. 17-26
5. Pyramid (Fig. 17-27): V 6. Cone (Fig. 17-28): V 7. Sphere (Fig. 17-29): V
1 3 Bh 1 3 Bh
or V
1 2 3 pr h
4 3 3 pr
r h B B h h r h r
Fig. 17-27
Fig. 17-28
Fig. 17-29
CHAPTER 17 Extending Plane Geometry into Solid Geometry
SOLVED PROBLEMS
Relations among cubic units Find the volume V of (a) A cubic foot in cubic inches (b) A cubic yard in cubic feet (c) A liter (cubic decimeter) in cubic centimeters
Solutions
(a) V e3 for a cube. Since 1 ft So 1 ft3 1728 in3. (b) V e3 for a cube. Since 1 yd So 1 yd3 27ft3. (c) V e3 again. Since 1 dm3 So 1 liter 1000 cm3. 12 in, V 3 ft, V 10 cm3, V 123 33 103 1728 27 1000
Finding volumes or cubes Find the volume V of a cube, in cubic feet, if one edge is (a) 4 in, (b) 4 ft, (c) 4 yd.
Solutions
To find the volume in cubic feet, we must express the side in feet. (a) V (b) V (c) V e3 and, since 4 in e3 43 64 ft3 12 ft, V 123 1728 ft3
1 3 ft,
A1 B 3
1 3 27 ft
e3, and since 4 yd
Finding the volumes of a rectangular solid, prism, and pyramid Find the volume of (a) A rectangular solid having a length of 6 in, a width of 4 in, and a height of 1 ft (b) A prism having a height of 15 yd and a triangular base of 120 ft2 (c) A pyramid having a height of 8 cm and a square base whose side is 41 cm 2
Solutions
(a) V (b) V (c) V lwh Bh
1 3 Bh
6(4)(12) 120(45)
1 9 2 3 2 (8)
288 in3 5400 ft3 54 cm3 200 yd3
Finding the volumes of a sphere, cylinder, and cone Find the volume of (a) A sphere with a radius of 10 in (b) A cylinder with a height of 4 yd and a base whose radius is 2 ft (c) A cone with a height of 2 ft and a base whose radius is 2 yd
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Solutions
In these calculations, we shall let p (a) V (b) V (c) V
4 3 3 pr 4 3 3 (3.14)10 2 3
4186 in3 150.72 ft3 75.36 ft3
pr2h
1 2 3 pr h
(3.14)(22)12
1 2 3 (3.14)(6 )(2)
Deriving formulas from V Bh From V Bh, the volume formula for a prism or cylinder, derive the volume formulas for the solids in Fig. 17-30.
Solutions
(a) Since B (b) Since B (c) Since B (d) Since B lw, V e and h
lwh e, V Bh (e2) e hr (b 2 e3 hhr (b 2
pr2, V Bh pr2h hr (b br), V Bh 2
br)h
h h B l (a) Rectangular solid w e (b) Cube e B e B r h B b h
(c) Cylinder of revolution
(d ) Right prism with a trapezoid for a base
Fig. 17-30
Formulas for command volumes State the formula for the volume of each solid in Fig. 17-31.
II b
b e e e e (a) e e e e a (b) a (c) e b c R = 3r
Fig. 17-31
CHAPTER 17 Extending Plane Geometry into Solid Geometry
Solutions
(a) V (b) V lwh for this solid. Now l lwh again. Here l Vcyl. I Vcyl. II 2a, w pR2h 4e, w 3e, and h 2e. Hence, V (4e)(3e)(2e) 6abc pr2h 10pr2h 24e3
c, and h
3b. Hence, V 3r, so V
(2a)(c)(3b) p(3r)2h
(c) Here V
pr2h. But R
SUPPLEMENTARY PROBLEMS
17.1. Find, to the nearest integer (using p (a) A cube with an edge of 7 yd (b) A rectangular solid with dimensions of 8 ft, 62ft, and 14 ft (c) A sphere with radius of 30 m (d) A cylinder of revolution with a radius of 10 yd and a height of 42 yd. [Hint: Use T
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