Volume by Calculation
The
volume of a rectangular solid is
equal to its length (l) times
width (w) times thickness (t) or
height.

Objective
To
state the relationship of length, width, and thickness to the volume
of a rectangular solid.
To
express a given volume in units of milliliters, cubic centimeters,
or cubic inches.
Have you ever wondered how a building contractor arrives at the
costs for a new home? For example, how does one estimate the cost
of pouring a foundation? The cost is based on the volume of concrete
that is poured. If the foundation is rectangular, the volume of
concrete is found by multiplying length (l) times width (w)
times thickness (t). We can express this relationship by
the following equation.
Before calculating a volume, we must express each of the measurements in the
same units. If we express length, width, and thickness in centimeters, the volume
of the solid has the units of cubic centimeters (symbol cm^{3}). For example, if a rectangular
solid measures 3 cm by 2 cm by 1 cm, we find the volume is 6 cm^{3} Example
Exercise 3.9 further illustrates how to determine the volume of a rectangular
solid by calculation.

Example Exercise 3.9


Let's now consider a rectangular solid with a length of 4 cm, a width of 3 cm,
and a volume of 24 cm^{3} If we are given length, width, and volume of a rectangular
solid, we can calculate its thickness. We obtain the thickness of the solid by dividing
its volume by its length and its width; that is, we divide 24 cm^{3} by 4 cm and
3 cm. In this example, the thickness of the solid is 2 cm. Example Exercise 3.10 further
illustrates finding the thickness of rectangular solid.

Example Exercise 3.10

One Liter
A liter equals the volume of a cube 10 cm on a side,
that is, 1000 cm3.
1Liter Bottle
A 1liter water bottle contains 1000 mL of liquid.
The Beetle
The original
Volkswagen Beetle was the
most widely sold automobile in
the world, and the newest
model is also proving popular.

CUBIC VOLUME AND LIQUID VOLUME
In the metric system, the basic unit of liquid volume is the liter. A liter is equivalent
to the volume occupied by a cube exactly 10 cm on a side.
A cube is a regular solid with its length, width, and thickness being equal. We
calculate the volume of the cube by multiplying length by width by thickness.
The calculated volume of 1 L is 1000 cm^{3}. Moreover, we recall the exact equivalent:
We can combine the two equivalents as follows.
Simplifying, we have
We see that 1 cm^{3} is exactly equivalent to 1 mL. Example Exercise 3.11 further illustrates
the calculation of volume.

