◆
Section 6–5
167
Volumes and Areas of Solids
17. A certain car tire is 78.5 cm in diameter. How far will the car move forward with one revolution of the wheel?
18. Archimedes claimed that the area of a circle is equal to the area of a triangle that has an
altitude equal to the radius of the circle and a base equal to the circumference of the circle.
Use the formulas of this chapter to show that this is true.
6–5
Volumes and Areas of Solids
Volume
The volume of a solid is a measure of the space it occupies or encloses. It is measured in cubic
units (m3, cm3, cu. ft., etc.) or, usually for liquids, in litres or gallons. One of our main tasks in
this chapter is to compute the volumes of various solids.
Area
We speak about three different kinds of areas. Surface area will be the total area of the surface
of a solid, including the ends, or bases. The lateral area, which will be defined later for each
solid, does not include the areas of the bases. The cross-sectional area is the area of the plane
figure obtained when we slice the solid in a specified way.
The formulas for the areas and volumes of some common solids are given in Fig. 6–60.
a
a
a
Base
122
Surface area = 6a 2
123
Cube
h
h
Volume = lwh
Rectangular
parallelepiped
l
w
Volume = a3
Surface area = 2(lw + hw + lh)
125
Any cylinder
or prism
Volume = (area of base)(altitude)
126
Right
cylinder
or prism
Lateral area = (perimeter of base)(altitude)
(not incl. bases)
127
Volume =
2r
4
πr 3
3
Any cone
or pyramid
h
h
s
A1
s
A2
h
Frustum
FIGURE 6–60
128
Sphere
Surface area = 4πr 2
s
124
Right circular
cone or
regular pyramid
Lateral area =
Any cone
or pyramid
Volume =
Right circular
cone or
Lateral area =
regular pyramid
Some solids.
h
(area
3
Volume =
s
(sum
2
s
2
129
of base)
130
(perimeter of base)
131
h
(A1
3
+ A2 + A1A2)
132
of base perimeters) = 2s (P1 + P2) 133
168
Chapter 6
Note that the formulas for the
cone and frustum of a cone are
the same as for the pyramid and
frustum of a pyramid.
◆
Geometry
◆◆◆ Example 17: Find the volume of a cone having a base area of 125 cm2 and an altitude of
11.2 cm.
Solution: By Eq. 130,
125(11.2)
volume 467 cm3
3
◆◆◆
Dimension
A geometric figure that has length but no area or volume (such as a line or a curve) is said to
be of one dimension. A geometric figure having area but not volume (such as a circle) is said to
have two dimensions, or be two dimensional. A figure having volume (such as a sphere) is said
to have three dimensions, or be three dimensional.
Exercise 5
◆
Volumes and Areas of Solids
Prism and Rectangular Parallelepiped
1. Find the volume of the triangular prism in Fig. 6–61.
2. A 2.00-cm cube of steel is placed in a surface grinding machine, and the
vertical feed is set so that 0.1 mm of metal is removed from the top of the cube
at each cut. How many cuts are needed to reduce the weight of the cube by 8.3 g?
772 mm
925 mm
30
6m
m
552
mm
FIGURE 6–61
For some of these problems, we
need Eq. A43: The weight (or
mass) equals the volume of the
solid times the density of the
material.
The density of iron varies between
7000 and 7900 kg/m3, depending
upon processing, and that of steel
varies between 7700 and 7900
kg/m3. We will use a density of
7800 kg/m3 (7.8 g/cm3) for both
iron and steel.
The engine displacement is the
total volume swept out by all of
the pistons.
3. A rectangular tank is being filled with liquid, with each cubic metre of liquid
increasing the depth by 2.0 cm. The length of the tank is 12.0 m.
(a) What is the width of the tank?
(b) How many cubic metres will be required to fill the tank to a depth of 3.0 m?
4. How many loads of gravel will be needed to cover 3.0 km of roadbed, 11 m wide,
to a depth of 7.5 cm if one truckload contains 8.0 m3 of gravel?
Cylinder
5. Find the volume of the cylinder in Fig. 6–62.
Area = 18.2 in.2
6. Find the volume and the lateral area of a right
circular cylinder having a base radius of 128 and a
11.2 in.
height of 285.
7. A 3.00-m-long piece of iron pipe has an outside
diameter of 7.50 cm and weighs 56.0 kg. Find the
wall thickness.
60°
8. A certain bushing is in the shape of a hollow cylinder 18.0 mm in diameter and 25.0 mm
long, with an axial hole 12.0 mm in diameter. If
Perimeter = 15.4 in.
the density of the material from which they are
FIGURE 6–62
made is 2.70 g/cm3, find the mass of 1000 bushings.
9. A steel gear is to be lightened by drilling holes through the gear. The gear is 8.90 cm thick.
Find the diameter d of the holes if each is to remove 340 g.
10. A certain gasoline engine has four cylinders, each with a bore of 82.0 mm and a piston
stroke of 95.0 mm. Find the engine displacement in litres.
Section 6–5
◆
Volumes and Areas of Solids
Cone and Pyramid
11. The circumference of the base of a right circular cone is 40.0 cm, and the slant height is
38.0 cm. What is the area of the lateral surface?
12. Find the volume of a circular cone whose altitude is 24.0 cm and whose base diameter is
30.0 cm.
_
13. Find the weight of the mast of a ship, its height being 20 m, the circumference at one end
1.5 m and at the other 1.0 m, if the density of the wood is 935 kg/m3.
14. The slant height of a right pyramid is 11.0 m, and the base is a 4.00-m square. Find the area
of the entire surface.
15. How many cubic decimetres are in a piece of timber 10.0 m long, one end being a 38.0-cm
square and the other a 30.0-cm square?
16. Find the total surface area and volume of a tapered steel roller 12.0 ft. long and having end
diameters of 12.0 in. and 15.0 in.
Sphere
17.
18.
19.
20.
Find the volume and the surface area of a sphere having a radius of 744.
Find the volume and the radius of a sphere having a surface area of 462.
Find the surface area and the radius of a sphere that has a volume of 5.88.
How many great-circle areas of a sphere would have the same area as that of the surface of
the sphere? Note: A great-circle area of a sphere is one that has the same diameter as the
sphere.
21. Find the weight in kilograms of 100 steel balls each 6.35 cm in diameter.
_
22. A spherical radome encloses a volume of 9000 m3. Assume that the sphere is complete.
(A radome, or radar dome, is a spherical protective enclosure for a microwave or radar antenna.)
(a) Find the radome radius, r.
(b) If the radome is constructed of a material weighing 2.00 kg/m2, find its weight.
Structures
23. A wheat silo complex in Saskatchewan consists of seven silos (Fig. 6–63). Each silo is a
cylinder with a cone on top and bottom. Each cylinder is 12 m tall, 4.0 m wide, and the
distance from the tip of the bottom cone to the tip of the top cone is 18 m.
(a) Find the volume of one silo.
(b) One railway grain car can hold 114 m3. How many cars would be needed to transport
the entire contents of seven full silos?
FIGURE 6–63
169
170
Chapter 6
◆
Geometry
Case Study Discussion—Engineers Without Borders
First, some unit conversion is needed:
The inside height of the container is 7 ft. 9 in., or 93 in., which, when converted to
millimetres, is 93 in. 25.4 mm/in. 2362.2, or 2360 mm (rounded down). Because
each panel is 80 mm thick, we will be able to stack 2360 mm/80 mm 29.5 panels. We
need to round this down to 29 panels high.
Next, the width of the container interior is 7 ft. 7 in., or 91 in, or 2310 mm rounded
(91 in. 25.4 mm/in. 2311.4 mm). Because the panels are 1030 mm wide with packaging, we can fit in 2310/1030 2.24, or 2 stacks width-wise.
Finally, the interior depth of the container is 19 ft. 3 in., or 231 in., or 5870 mm
rounded (231 in. 25.4 mm/in. 5867.4). Because each panel is 1689 mm long with
packaging, we can fit in 5870/1689 3.48, or 3 rows deep. So the container can hold
6 vertical stacks of 29 panels, fitted 2 stacks across and 3 stacks deep, for a total of
174 panels (3 2 29 174).
◆◆◆
CHAPTER 6 REVIEW PROBLEMS ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
1. A rocket ascends at an angle of 60.0 with the horizontal. After 1.00 min it is directly over
a point that is a horizontal distance of 12.0 km from the launch point. Find the speed of the
rocket.
2. A rectangular beam 32 cm thick is cut from a log 40 cm in diameter. Find the greatest
depth beam that can be obtained.
3. A cylindrical tank 4.00 m in diameter is placed with its axis vertical and is partially filled
with water. A spherical diving bell is completely immersed in the tank, causing the water
level to rise 1.00 m. Find the diameter of the diving bell.
_
4. Two vertical piers are 240 ft. apart and support a circular bridge arch. The highest point of
the arch is 30.0 ft. higher than the piers. Find the radius of the arch.
5. Two antenna masts are 10 m and 15 m high and are 12 m apart. How long a wire is needed
to connect the tops of the two masts?
_
6. Find the area and the side of a rhombus whose diagonals are 100 cm and 140 cm.
7. Find the area of a triangle that has sides of length 573, 638, and 972.
8. Two concentric circles have radii of 5.00 and 12.0. Find the length of a chord of the larger
circle, which is tangent to the smaller circle.
9. A belt that does not cross goes around two pulleys, each with a radius of 10.0 cm and
whose centres are 22.0 cm apart. Find the length of the belt.
10. A regular triangular pyramid has an altitude of 12.0 m, and the base is 4.00 m on a side.
Find the area of a section made by a plane parallel to the base and 4.00 m from the vertex.
11. A fence parallel to one side of a triangular field cuts a second side into segments of 15.0
m and 21.0 m long. The length of the third side is 42.0 m. Find the length of the shorter
segment of the third side.
12. When Figure 6–64 is rotated about axis AB, we generate a cone inscribed in a hemisphere
which is itself inscribed in a cylinder. Show that the volumes of these three solids are in the
ratio 1 : 2 : 3. (Archimedes was so pleased with this discovery, it is said, that he ordered this
figure to be engraved on his tomb.)
13. Four interior angles of a certain irregular pentagon are 38, 96, 112, and 133. Find the
fifth interior angle.
14. Find the area of a trapezoid whose bases have lengths of 837 m and 583 m and are separated by a distance of 746 m.