Calculate the Volume of a Prism - dummies

Calculate the Volume of a Prism

By Mark Ryan

To calculate the volume of a prism, you first need to know its height and the area of its base (top or bottom, it doesn’t matter—they’re parallel and congruent).

geometry-prism
A prism with its bases and lateral rectangles.

The volume of a prism is given by the following formula:

geometry-prism-volumne

An ordinary box is a special case of a prism, so you can use this volume formula for a box, but you probably already know the easier way to compute a box’s volume:

geometry-box-volume

(Because the length times the width gives you the area of the base, these two methods really amount to the same thing.) To get the volume of a cube, the simplest type of box, you just take the length of one of its edges and raise it to the third power

geometry-volume-cube

Now, take a look at the formula in action.

geometry-prism-formula

Here’s the diagram.

geometry-prism-diagram

To use the volume formula, you need the prism’s height (line CD) and the area of its base (triangle AED). (You’ve probably noticed that this prism is lying on its side. That’s why its height isn’t vertical and its base isn’t on the bottom.)

Get the height first. ABCD is a square, so triangle BCD (half of the square) is a 45°-45°-90° triangle with a hypotenuse of 8. To get the leg of a 45°-45°-90° triangle, you divide the hypotenuse by the square root of 2 (or use the Pythagorean Theorem, noting that a = b in this case). So that gives you

geometry-line-cd

for the length of line CD, which, again, is the height of the prism.

And here’s how you get the area of triangle AED. First, note that AD, like CD, is

geometry-triangle-aed

Next, because angles EAD and EDA are given 45-degree angles, angle AED must be 90 degrees; thus, triangle AED is another 45°-45°-90° triangle.

geometry-angle-aed

The area of a right triangle is given by half the product of its legs (because you can use one leg for the triangle’s base and the other for its height), so

geometry-right-triangle

You’re all set to finish with the volume formula:

geometry-prism-finish