How To Find The Volume Of A Square Pyramid

The volume of an object is defined as the three-dimensional space it occupies, but it might be easier to think of it as the amount of water, gas or any other substance that said object would hold. Either way, when faced with a square-based pyramid – think the pyramids of Egypt as an example – you can find its volume using a simple formula that requires the height of the pyramid and the length of one side along its base.

1. Collect the Essential Information

Collect, measure or calculate the height of the pyramid and the length of one side along its base. Consider the example of a square pyramid where one side of the pyramid's base measures 5 inches, and the height of the pyramid is 6 inches.

Both measurements must be made in the same units. Also, to use this formula, the height must be the distance from the uppermost vertex of the pyramid (its peak) straight down to the middle of the base, not the slant height from the peak of the pyramid to one of its lower vertexes.

If you're given the slant height of the pyramid, it represents the hypotenuse of a right triangle formed by itself, the height of the pyramid, and 1/2 the length of the pyramid's base. Use the Pythagorean theorem:

\(a^2 + b^2 = c^2\)

to find the height of the pyramid. In this case c is the slant height of the pyramid, a is 1/2 the length of the base, and b will be the height of the pyramid.

2. Find the Area of the Base

Square the length of the pyramid's base or, in other words, multiply the length by itself. This gives you the area of the pyramid's base in square units. To continue the example, this would be:

\(5 \text{ inches} × 5 \text{ inches} = 25 \text{ inches}^2\)

3. Multiply by h/3

Multiply the area of the pyramid's base by the pyramid's height, then divide the answer by 3. The result is the volume of your pyramid, written in units cubed. To continue the example, you have:

\(25 \text{ inches}^2 × 6 \text{ inches}= 150 \text{ inches}^3\)

Divide this by three to get the pyramid's volume:

\(150 \text{ inches}^3 ÷ 3 = 50 \text{ inches}^3\)