What this calculator does
A square pyramid, the shape of the Great Pyramid of Giza, has a square base and four triangular faces meeting at a single apex directly above the base’s centre. It is the special, most commonly encountered case of a pyramid where all four base sides are equal, rather than the general rectangular base of a wider pyramid calculator.
This calculator takes that one base side length together with the height and returns the square pyramid volume directly, with the base area, the slant height and the lateral edge length shown alongside for the same figure.
The formula
Like any pyramid, volume is one third of the prism sharing the same base and height: V = ⅓ × a² × h, where a² is the square base area. The slant height, the distance from the apex down the middle of a triangular face to the base edge, follows from Pythagoras as l = √(h² + (a/2)²), using half the base side and the vertical height.
| Term | Meaning |
|---|---|
| a | The base side length. All four sides of a square base are equal. |
| h | The vertical height, from the centre of the base straight up to the apex. |
| Slant height | The distance from the apex down the middle of one triangular face to the base edge, shorter than the lateral edge. |
The inputs explained
| Field | What to enter |
|---|---|
| Base side | The length of one side of the square base. |
| Height | The vertical height, from the centre of the base straight up to the apex. |
When to use it
Volume of a square pyramid monument or roof
A pyramid-shaped monument, skylight or roof section with a square footprint has its enclosed volume from just the base side and the height, the two measurements usually already on a drawing.
Volume of a square based pyramid stockpile
A pile of sand, grain or aggregate that has settled into a roughly square-based pyramid shape gets a workable volume estimate from a base side and a height, without needing separate length and width measurements.
Checking a volume of square pyramid homework answer
A geometry problem giving a base side and height is solved directly here, without separately squaring the base, multiplying by the height, and dividing by three by hand.
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.
Volume of a square pyramid across a range of heights
A fixed base side of 10, across a range of heights from short to tall.
| Height | Volume | Slant height (apex to mid-edge) |
|---|---|---|
| 6 | 200.000 | 7.810 |
| 9 | 300.000 | 10.296 |
| 12 | 400.000 | 13.000 |
| 15 | 500.000 | 15.811 |
| 18 | 600.000 | 18.682 |
Volume of a square pyramid across a range of base sides
A fixed height of 12, across a range of base side lengths.
| Base side | Volume | Base area |
|---|---|---|
| 4 | 64.000 | 16.000 |
| 8 | 256.000 | 64.000 |
| 12 | 576.000 | 144.000 |
| 16 | 1,024.00 | 256.000 |
| 20 | 1,600.00 | 400.000 |
Questions
What is the formula for the volume of a square pyramid?
Volume = ⅓ × a² × h, where a is the base side length and h is the vertical height. It is the same one-third rule that applies to any pyramid or cone, using a square base area.
What is the volume of a square based pyramid with a base of 6 and a height of 9?
Volume = ⅓ × 6² × 9 = ⅓ × 36 × 9 = 108. Entering those two values here confirms the same figure alongside the base area and slant height.
Is the slant height the same as the height?
No. The height is the straight vertical distance from the base centre to the apex, used directly in the volume formula. The slant height runs down the middle of a sloped triangular face instead, and is always longer than the vertical height.
How is a square pyramid different from a general pyramid calculator?
A square pyramid has one base side length, since all four sides match. A general rectangular pyramid needs a separate length and width, and usually has two different slant heights rather than one, since its two pairs of triangular faces are not identical.
For the surface area of the same shape, see the surface area of a square pyramid calculator. For a pyramid with a rectangular, non-square base, see the general pyramid calculator.