Pyramid volume is one of 8 solids formulas in the geometry section of this library, and it is used at middle school level.
Why pyramid volume works
Three identical pyramids sharing a base and a height can be fitted together to fill a prism built on that same base, with no space left over. Each one therefore takes a third of the prism, and that same third shows up again for cones.
What each symbol means
$B$ is base area and $h$ is perpendicular height.
Pyramid volume: when it holds
Use perpendicular height, not slant height.
When it stops applying
It assumes a single apex above one base. A frustum, which is a pyramid with its top sliced off, is not covered, so you must subtract the small missing pyramid from the full one or use a frustum formula.
Pyramid volume: a worked example
Base area $36$ and height $10$ give $V=120$.
The mistake to avoid
What people do: Using the slant height, the distance measured up the middle of a triangular face.
Why it goes wrong: The slant height is the hypotenuse of a right triangle whose other leg is the true height, so it is always longer and the volume comes out too big.
Do this instead: Use the straight vertical drop from the apex down to the base plane, recovering it from the slant height with the Pythagorean theorem if necessary.
Pyramid volume: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $B$ is base area and $h$ is perpendicular height.
- Check the conditions before substituting. Use perpendicular height, not slant height.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Solids
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where pyramid volume comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about pyramid volume
Why one third rather than one half?
Because three copies, not two, are needed to fill the matching prism. The same one third appears in the cone rule for the same reason.
Does the apex have to sit above the centre of the base?
No. Sliding the apex sideways keeps both the base and the height the same, so the volume does not change at all.
How do I get the height from the slant height?
The height, half the base side, and the slant height form a right triangle, so square the slant height, subtract, and take the root.
Does the base have to be a square?
No. Any flat polygon works, and the base area is simply that polygon's area, whether it is a triangle, a rectangle, or a hexagon.