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 cm3). For example, if a rectangular solid measures 3 cm by 2 cm by 1 cm, we find the volume is 6 cm3 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 cm3 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 cm3 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.

 



  1-Liter Bottle
A 1-liter 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 cm3. Moreover, we recall the exact equivalent: We can combine the two equivalents as follows.

Simplifying, we have

We see that 1 cm3 is exactly equivalent to 1 mL. Example Exercise 3.11 further illustrates the calculation of volume.

Example Exercise 3.11