Volume of a Right Rectangular Prism

Volume is a measure of how much space a three-dimensional figure occupies. In everyday language, the volume tells you how much a three-dimensional figure can hold. The basic unit of volume is the cubic unit—cubic centimeter, cubic inch, cubic meter, cubic foot, and so on. Each basic cubic unit has a measure of 1 for its length, width, and height.

Two postulates that apply to volume are the Volume Congruence Postulate and the Volume Addition Postulate.

Volume Congruence Postulate

If two polyhedrons (or solids) are congruent, then their volumes are congruent.

Volume Addition Postulate

The volume of a solid is the sum of the volumes of all of its non-overlapping parts.

A right rectangular prism is a prism with rectangular bases and the angle between each base and its rectangular lateral sides is also a right angle. You can recognize a right rectangular prism by its “box” shape.

You can use the Volume Addition Postulate to find the volume of a right rectangular prism by counting boxes. The box below measures 2\;\mathrm{units} in height, 4\;\mathrm{units} in width, and 3\;\mathrm{units} in depth. Each layer has 2 \times 4 \;\mathrm{cubes} or 8\;\mathrm{cubes}.

Together, you get three groups of 2\cdot4 so the total volume is:

V &= 2 \cdot 4 \cdot 3\\ &= 24

The volume is 24 \;\mathrm{cubic\ units}.

This same pattern holds for any right rectangular prism. Volume is giving by the formula:

\text{Volume} = l \cdot w \cdot h

Example 11

Find the volume of this box.

Use the formula for volume of a right rectangular prism.

V &= l \cdot w \cdot h \\ V &= 8 \cdot 10 \cdot 7 \\ V &=560

So the volume of this rectangular prism is 560\;\mathrm{cubic\ units}.

Volume of a Right Prism

Looking at the volume of right prisms with the same height and different bases you can see a pattern. The computed area of each base is given below. The height of all three solids is the same, 10.

Putting the data for each solid into a table, we get:

Solid Height Area of base Volume
Box 10 300 3000
Trapezoid 10 140 1400
Triangle 10 170 1700

The relationship in each case is clear. This relationship can be proved to establish the following formula for any right prism:

Volume of a Right Prism

The volume of a right prism is V = Bh.

where B is the area of the base of the three-dimensional figure, and h is the prism’s height (also called altitude).

Example 12

Find the volume of the prism with a triangular equilateral base and the dimensions shown in centimeters.

To find the volume, first find the area of the base. It is given by:

A = \frac{1}{2} bh

The height (altitude) of the triangle is 10.38 \;\mathrm{cm}. Each side of the triangle measures 12 \;\mathrm{cm}. So the triangle has the following area.

A &= \frac{1}{2} bh \\ &= \frac{1}{2} (10.38)(12) \\ &= 62.28

Now use the formula for the volume of the prism, V=Bh, where B is the area of the base (i.e., the area of the triangle) and h is the height of the prism. Recall that the "height" of the prism is the distance between the bases, so in this case the height of the prism is 15 \;\mathrm{cm}. You can imagine that the prism is lying on its side.

V &= Bh \\ &= (62.28)(15) \\ &= 934.2

Thus, the volume of the prism is 934.2 \;\mathrm{cm}^3.

Example 13

Find the volume of the prism with a regular hexagon for a base and 9-\mathrm{inch} sides.

You don’t know the apothem of the figure’s base. However, you do know that a regular hexagon is divided into six congruent equilateral triangles.

You can use the Pythagorean Theorem to find the apothem. The right triangle measures 9 by 4.5 by a, the apothem.

9^2 &= 4.5^2 + n^2\\ 81 &= 20.25 + n^2\\ 60.75 &= n^2\\ 7.785 &= n

A &= \frac{1}{2} \text{(apothem)}\ \text{(perimeter)}\\ &= \frac{1}{2} (7.785)(6 \cdot 9) \\ &= 210.195 \ \text{sq in}

Thus, the volume of the prism is given by:

V &= Bh \\ &= 210.195 \cdot 24 \\ &= 5044.7 \ \text{cu in}

Last modified: Wednesday, June 30, 2010, 11:27 AM