Pyramid Volume Calculator
Find the volume of a square or rectangular pyramid from its base dimensions and height, with base area and surface area.
How it works
Volume = ⅓ · base area · h · Base area = l · w · Square pyramid slant = √(h² + (l ÷ 2)²) · SA = base + 2 · l · slant
Like a cone, a pyramid occupies exactly one third of the prism that shares its base and height, regardless of the base shape. So the volume is one third of the base area times the vertical height. For a square pyramid, each triangular face rises along a slant height √(h² + (l ÷ 2)²) measured to the midpoint of a base edge, and the four faces plus the base give the total surface area.
Worked example
A square pyramid with base 6 × 6 and height 9: base area = 36, so volume = ⅓ × 36 × 9 = 108 cubic units. The slant height is √(9² + 3²) = √90 ≈ 9.4868, the four faces add 2 × 6 × 9.4868 ≈ 113.842, and the total surface area is 36 + 113.842 ≈ 149.842.
Sources: Pyramid (geometry) — Wikipedia
Frequently asked questions
Why one third of the base times height?
Integrating the shrinking cross-sections from base to apex — each a scaled copy of the base — yields exactly ⅓ × base area × height. Three identical pyramids can be assembled into one prism of the same base and height, showing the factor geometrically.
What is the slant height versus the height?
The height h is the straight vertical drop from the apex to the base centre. The slant height runs down the middle of a triangular face to a base edge, equal to √(h² + (l ÷ 2)²), and is used for the face areas.
Does the surface area formula work for a rectangular pyramid?
A rectangular pyramid has two different slant heights, one per pair of faces, so this calculator reports the surface area only for the square case. The volume ⅓ × l × w × h is correct for both.
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This calculator is for educational purposes. Double-check important results before acting on them.