Volume of a Pyramid — Formula & Examples

To find the volume of a pyramid, use $V = \frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the perpendicular height. This works for square, rectangular, and triangular pyramids because the formula depends on the base area, not the base shape.

$$V = \frac{1}{3}Bh$$

If you already know the base area, the problem is usually quick: multiply by the perpendicular height, then divide by $3$.

What the pyramid volume formula means

The factor $B$ stands for the whole area of the base, not just one side length. That matters because pyramids can have different base shapes, such as square, rectangular, or triangular bases.

The factor $h$ is the straight height from the base to the apex. It must be perpendicular to the base. If a problem gives a slant height, that is not enough by itself unless you first convert it to the perpendicular height.

The factor $\frac{1}{3}$ tells you a pyramid has one-third the volume of a prism with the same base area and the same perpendicular height.

Why there is a one-third

A prism with the same base area $B$ and height $h$ has volume

$$V_{\text{prism}} = Bh$$

A pyramid tapers to a point, so it holds less space than that prism. For the same base area and perpendicular height, it holds exactly one-third as much:

$$V_{\text{pyramid}} = \frac{1}{3}Bh$$

That comparison is often the easiest way to remember the formula.

Worked example: volume of a square pyramid

Suppose a square pyramid has base side length $6$ cm and perpendicular height $10$ cm.

First find the base area:

$$B = 6^2 = 36 \text{ cm}^2$$

Then substitute into the formula:

$$V = \frac{1}{3}Bh$$ $$V = \frac{1}{3}(36)(10)$$

Now simplify:

$$V = \frac{360}{3} = 120$$

So the volume is $120 \text{ cm}^3$.

The units are cubic centimeters because volume measures three-dimensional space.

Square pyramid and rectangular pyramid formulas

The main formula stays the same. What changes is how you compute the base area.

For a square pyramid with base side length $s$,

$$B = s^2$$

so

$$V = \frac{1}{3}s^2 h$$

For a rectangular pyramid with base length $l$ and width $w$,

$$B = lw$$

so

$$V = \frac{1}{3}lwh$$

Both are just special cases of $V = \frac{1}{3}Bh$.

Common mistakes when finding the volume of a pyramid

1. Using a side length instead of the full base area. The formula needs $B$, not just one measurement from the base.
2. Using slant height instead of perpendicular height. Volume depends on the straight height to the base.
3. Forgetting the $\frac{1}{3}$. Without it, you are calculating the matching prism volume.
4. Writing square units instead of cubic units. Base area uses square units, but volume uses cubic units.
5. Using a shortcut such as $V = \frac{1}{3}s^2 h$ when the base is not actually a square.

When this formula is used

This formula appears in geometry, mensuration, construction estimates, and any setting where a solid can be modeled as a pyramid or near-pyramid.

If the object is only approximately pyramidal, the result is also an estimate. The formula is still useful when you need a practical volume quickly.

Quick check before you finish

If a prism with the same base area and height would have volume $Bh$, then the pyramid should have volume $Bh/3$.

So if your answer comes out equal to $Bh$, you have probably missed the factor $\frac{1}{3}$.

Try a similar problem

Try a rectangular pyramid with base $8$ cm by $5$ cm and perpendicular height $9$ cm. Find the base area first, then apply $V = \frac{1}{3}Bh$.

If you want to try your own numbers and compare cases quickly, solve a similar volume problem with GPAI Solver and check how the answer changes when only the base area or height changes.