What this calculator does
A rectangular pyramid has a rectangular base, with length and width that are not necessarily equal, tapering up to a single apex. Its volume follows the same one-third rule as any pyramid or cone: it holds exactly a third of the volume of a rectangular box built from the same base and height.
The formula for the volume of a rectangular pyramid is one-third times length times width times height. Where a square pyramid uses one base measurement squared, a rectangular pyramid needs two separate base dimensions, since the base is not a square.
The formula
Multiply the base length by the base width to get the base area, then multiply by the height and divide by three. The result is the volume the pyramid encloses, in cubic units matching whatever unit the length, width and height were measured in.
| Term | Meaning |
|---|---|
| V | Volume, in cubic units. |
| Base length and width | The two side lengths of the rectangular base, measured at right angles to each other. |
| Height | The perpendicular distance from the base to the apex, not the slanted edge length. |
The inputs explained
| Field | What to enter |
|---|---|
| Base length | The longer side of the rectangular base. |
| Base width | The shorter side of the rectangular base. |
| Height | The vertical height from the base plane up to the apex, measured straight up, not along a sloped face. |
When to use it
A rectangular-based roof or canopy
A pyramid-shaped roof section, skylight or canopy over a rectangular room is not usually built on a square footprint, so the volume of enclosed space needs both base dimensions rather than one.
Estimating material for a pyramid-shaped mould or hopper
A hopper, funnel top or decorative pyramid form built on a rectangular base uses this same formula to work out how much material fills it.
A maths or engineering exercise with a non-square base
Once a pyramid problem specifies two different base dimensions, the general rectangular-pyramid formula applies rather than the simpler square-base version.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume changes as base length increases at a fixed width and height
A fixed base width of 8 and height of 10, across a range of base lengths.
| Base length | Volume | Base area |
|---|---|---|
| 6 | 160.000 | 48.000 |
| 8 | 213.333 | 64.000 |
| 10 | 266.667 | 80.000 |
| 12 | 320.000 | 96.000 |
| 15 | 400.000 | 120.000 |
| 20 | 533.333 | 160.000 |
How volume changes with height at a fixed base
A fixed base of 12 by 8, across a range of heights.
| Height | Volume | Base area |
|---|---|---|
| 4 | 128.000 | 96.000 |
| 6 | 192.000 | 96.000 |
| 8 | 256.000 | 96.000 |
| 10 | 320.000 | 96.000 |
| 15 | 480.000 | 96.000 |
| 20 | 640.000 | 96.000 |
Questions
How is this different from a square pyramid?
A square pyramid is the special case where the base length and width are equal, so its volume formula is one-third times side squared times height. This calculator handles the general case where the two base sides differ, using length times width instead of side squared.
Why divide by three?
Any pyramid or cone occupies exactly one third of the volume of the prism or cylinder that shares its base and height. That factor of one third comes from integrating the shrinking cross-section from base to apex, and it holds regardless of the base shape.
Does the height have to be measured straight up?
Yes. The height in this formula is the perpendicular distance from the base plane to the apex, not the length of a sloped edge or face. Using a slant length instead of the true height will overstate the volume.
What if I only know the slant height or edge length?
You would need to work back to the true perpendicular height first, typically using the Pythagorean theorem with the slant measurement and half the relevant base dimension, before this formula applies.
For a pyramid with an equal-sided square base, see the square pyramid volume calculator. For the volume of a rectangular box rather than a pyramid, see the volume of rectangle calculator.