Find the volume of a pyramid with a square, rectangular, triangular, or regular-polygon base — or enter a known base area. Uses V = (1/3) × B × h, with optional slant-height input for square bases and multi-unit results.
V = (1/3) × B × h
B = base area · h = perpendicular height from base to apex
Side length of the square base
Vertical distance from base to apex
Every pyramid with a flat base has volume equal to one third of a prism with the same base and height. That single rule — V = (1/3) × B × h — is the pyramid volume formula taught in most US and UK school courses. Compute the base area B for your shape, multiply by the perpendicular height h from the base plane to the apex, then take one third. Keep every length in the same unit so the answer appears in that unit cubed.
V = (1/3) × B × h Square base: B = a² Rectangular base: B = l · w Triangular base: B = ½ · b · t Regular n-gon: B = n · s² / (4 · tan(π/n)) Square with slant height ℓ: h = √(ℓ² − (a/2)²) (requires ℓ > a/2)
B = 6² = 36. V = (1/3) × 36 × 9 = 108 cubic units.
B = 4 × 5 = 20. V = (1/3) × 20 × 6 = 40 cubic units.
B = ½ × 6 × 4 = 12. V = (1/3) × 12 × 10 = 40 cubic units.
h = √(13² − 5²) = √(169 − 25) = √144 = 12. B = 100. V = (1/3) × 100 × 12 = 400 cubic units.
A prism with base area B and height h has volume B × h. Three congruent pyramids with that same base and height can fill the prism completely (a standard solid-geometry result, often linked to Cavalieri’s principle). Therefore the pyramid takes exactly one third of the prism volume: Vpyramid = (1/3) × B × h. The factor does not change when the base is a triangle, square, rectangle, or regular polygon — only B changes with the base shape.
This is also why a cone uses the same one-third factor with a circular base (B = πr²). If you ever forget the pyramid rule, remember “same base, same height, one third of the prism.” That mental check catches the most common homework mistake: forgetting the 1/3 and reporting the prism volume instead.
| Base | Base area B | Volume |
|---|---|---|
| Square (edge a) | a² | (1/3)·a²·h |
| Rectangular (l × w) | l·w | (1/3)·l·w·h |
| Triangular (b, t) | ½·b·t | (1/3)·(½·b·t)·h |
| Regular n-gon (side s) | n·s²/(4·tan(π/n)) | (1/3)·B·h |
| Known base area | B (given) | (1/3)·B·h |
Exam questions sometimes give the face slant height ℓ (from the midpoint of a base edge up the triangular face to the apex) instead of the vertical height h. On a square base of edge a, a right triangle with legs a/2 and h has hypotenuse ℓ, so h = √(ℓ² − (a/2)²). The option “I know the slant height” applies that conversion automatically. If ℓ ≤ a/2 the expression under the square root is not positive and the geometry is impossible — for example a = 10 with slant 5 fails because a/2 = 5.
Lateral and total surface area (square / rectangular bases) use the face slant heights from vertical h. For a square face slant ℓface = √(h² + (a/2)²), lateral area = 2·a·ℓface. See also the Square Pyramid Surface Area Calculator.
For any pyramid with a flat base, V = (1/3) × B × h, where B is the base area and h is the perpendicular height from the base to the apex. That is exactly one third of a prism with the same base and height.
Three congruent pyramids with the same base and height fill a prism of that base and height (Cavalieri’s principle / standard solid-geometry result). So the pyramid takes one third of the prism volume: V = (1/3) × B × h.
The volume formula needs the perpendicular (vertical) height h. If you only know the face slant height ℓ on a square pyramid, convert with h = √(ℓ² − (a/2)²), provided ℓ > a/2.
Yes. Compute the base area B for your shape (½·b·t for a triangle, or n·s²/(4·tan(π/n)) for a regular n-gon), then apply V = (1/3) × B × h.
For square and rectangular bases this calculator also reports lateral and total surface area from the vertical height. For a dedicated slant-height workflow, use the Square Pyramid Surface Area calculator.
Square Pyramid Volume Calculator: The (1/3) × s² × h Formula
Finding the volume of a square-based pyramid is a common geometry task in both US and UK classrooms. The rule is easy to remember once you link it to prisms: a pyramid with the same base and height fills exactly one third of the matching prism. Use the free Pyramid Volume Calculator (linked just below) for square, rectangular, triangular and polygon bases — including slant-height input — then come back here for the worked explanation.
Start with the area of the square base, A = s². The volume is one third of what you would get if you stretched that base through the full height of a prism: V = (1/3) × A × h. The height h must be measured straight up from the base to the apex, not along a slanted edge on a side face. If you mix centimetres and metres, convert first so every length uses the same unit; the volume then appears in the matching cubic unit.
V = (1/3) × s² × h, where s is the base side and h is the perpendicular height. Equivalently, V = (1/3) × (base area) × h.
Take s = 3 cm and h = 12 cm. The base area is 9 cm². Multiply by the height to get 108 cm³ for a prism match, then take one third: V = 36 cm³. You can picture three identical pyramids filling the same box as one prism with the same base and height.
Open the Pyramid Volume Calculator, pick your base shape (square, rectangular, triangular, regular polygon, or known base area), enter the dimensions in one length unit, then Calculate. For a square base you can tick “I know the slant height” instead of entering vertical h. Read base area, volume in several cubic units, and the step-by-step working. If you see an error, check positives and that slant height exceeds a/2.
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