Chapter 27

Volumes of common solids

27.1 Introduction

Thevolumeof any solid is a measure of the space occu-
pied by the solid. Volume is measured incubic units
such as mm^3 ,cm^3 and m^3.
This chapter deals with finding volumes of common
solids; in engineering it is often important to be able to
calculatevolumeorcapacitytoestimate,say,theamount
of liquid, such as water, oil or petrol, in different shaped
containers.
Aprismis a solid with a constant cross-section and
with two ends parallel. The shape of the end is used to
describe the prism. For example, there are rectangular
prisms (called cuboids), triangular prisms and circular
prisms (called cylinders).
On completing this chapter you will be able to cal-
culate the volumes and surface areas of rectangular and
other prisms, cylinders, pyramids, cones and spheres,
together with frusta of pyramids and cones. Volumes of
similar shapes are also considered.

27.2 Volumes and surface areas of

common shapes

27.2.1 Cuboids or rectangular prisms

A cuboid is a solid figure bounded by six rectangular
faces; all angles are right angles andopposite faces are
equal. A typical cuboid is shown in Figure 27.1 with
lengthl, breadthband heighth.

Volume of cuboid=l×b×h

and

surface area= 2 bh+ 2 hl+ 2 lb= 2 (bh+hl+lb)

h

b

l

Figure 27.1

Acubeis a square prism. If all the sides of a cube arex

then

Volume=x^3 and surface area= 6 x^2

Problem 1. A cuboid has dimensions of 12cm by

4cm by 3cm. Determine (a) its volume and (b) its

total surface area

The cuboid is similar to that in Figure 27.1, with

l=12cm,b=4cmandh=3cm.

(a) Volume of cuboid=l×b×h= 12 × 4 × 3

=144cm^3

(b) Surface area= 2 (bh+hl+lb)

= 2 ( 4 × 3 + 3 × 12 + 12 × 4 )

= 2 ( 12 + 36 + 48 )

= 2 × 96 =192cm^2

Problem 2. An oil tank is the shape of a cube,

each edge being of length 1.5m. Determine (a) the

maximum capacity of the tank in m^3 and litres and

(b) its total surface area ignoring input and output

orifices

DOI: 10.1016/B978-1-85617-697-2.00027-2