Volume of a Pyramid — Formula & Examples

Use $V = \frac{1}{3}Bh$ whenever a solid rises from a flat base to a single apex: a square pyramid, a rectangular pyramid, a triangular pyramid, or any near-pyramid in construction and mensuration problems. Because the formula depends on the base area $B$, not the base shape, the same procedure covers every one of them.

Here $B$ is the area of the base and $h$ is the perpendicular height. If you already know the base area, the work is short: multiply by the perpendicular height, then divide by $3$.

The Procedure, Step By Step

1. Find the base area first and call it $B$.
2. Identify the perpendicular height, the straight height from base to apex, not a slanted measurement.
3. Substitute into $V = \frac{1}{3}Bh$, multiplying base area by height and dividing by $3$.
4. Check the units. If the base dimensions and height share a length unit, the volume comes out in cubic units.

Each piece carries meaning. $B$ is the whole area of the base, not a single side length, which is what lets the formula handle square, rectangular, and triangular bases alike. The height $h$ must be perpendicular to the base; a slant height is not enough until you convert it. And the $\frac{1}{3}$ says a pyramid holds one-third the volume of a prism with the same base area and perpendicular height.

Why The One-Third Appears

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

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

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

A Full Example: Square Pyramid

Take a square pyramid with base side length $6$ cm and perpendicular height $10$ cm.

Find the base area first:

Substitute into the formula:

Simplify:

So the volume is $120 \text{ cm}^3$. The units are cubic centimeters because volume measures three-dimensional space.

The only thing that changes for other bases is step 1. For a square base of side $s$, $B = s^2$, so $V = \frac{1}{3}s^2 h$. For a rectangular base of length $l$ and width $w$, $B = lw$, so $V = \frac{1}{3}lwh$. Both are just $V = \frac{1}{3}Bh$ with the base area filled in.

Where Each Step Goes Wrong

• Finding the base area: using a single side length instead of the full area $B$.
• Identifying the height: using slant height instead of the perpendicular height to the base.
• Substituting: forgetting the $\frac{1}{3}$, which leaves you with the matching prism volume.
• Checking units: writing square units when volume needs cubic units.
• A shortcut trap: applying $V = \frac{1}{3}s^2 h$ when the base is not actually a square.

A self-check before you finish: a prism with the same base area and height would have volume $Bh$, so the pyramid should come out to $Bh/3$. If your answer equals $Bh$, you have probably missed the $\frac{1}{3}$. And note the model condition: if the object is only approximately pyramidal, the result is an estimate, still useful for a quick practical volume.

Run It Yourself

Try a rectangular pyramid with base $8$ cm by $5$ cm and perpendicular height $9$ cm. Find the base area first, then apply $V = \frac{1}{3}Bh$. To test your own numbers and watch how the answer responds when only the base area or height changes, solve a similar volume problem with GPAI Solver.