Pyramid Volume Calculator

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Key Features

Instant Calculation

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A pyramid's volume is always exactly one-third of its base area times its height, regardless of what shape the base is — triangular, square, pentagonal, or any other polygon. This "1/3" relationship isn't a coincidence: it can be proven rigorously with calculus (integrating cross-sectional area from the apex down to the base) or demonstrated physically, since three identical pyramids with matching base and height can be assembled to exactly fill a prism of the same base and height. Architects and engineers use this formula for anything from estimating the concrete needed for a pyramid-shaped foundation to calculating the capacity of a pyramidal hopper or funnel in industrial equipment.

How it works

Pick "any base" and enter the base area directly (useful once you've already computed the area of a triangular, pentagonal, or otherwise irregular base), or pick "square/rectangular base" and enter length and width so the calculator finds the base area for you. Either way, also enter the height — measured perpendicular from the base up to the apex, not the slant height along a face. The calculator then applies V = (1/3) × base area × height.

Enter base type.
Enter base area (general mode).
Enter base length (rectangular mode).
Enter base width (rectangular mode).
Enter height.
Click Calculate to see your results.

Examples

Base area = 30, height = 9

V = (1/3)(30)(9) = 90 cubic units.

Square base, length = width = 6, height = 10

Base area = 6×6 = 36. V = (1/3)(36)(10) = 120 cubic units.

Common mistakes to avoid

Forgetting the 1/3 factor and computing base area × height instead of one-third of that.
Using the slant height instead of the true perpendicular height.
Computing the base area incorrectly for a non-rectangular base when using "any base" mode.

Frequently asked questions

Yes — compute the triangular base's area separately (for example, with Heron's formula) and enter it in the "any base" mode; the 1/3 × base × height relationship holds regardless of the base's shape.

This can be shown with calculus by integrating the pyramid's cross-sectional area (which shrinks proportionally to the square of the distance from the apex) from the apex to the base — the result always works out to exactly one-third, independent of the base's specific shape.

No — the height in this formula is the perpendicular (vertical) distance from the apex straight down to the base plane, not the slant height measured along a triangular face. Using slant height by mistake will give an incorrect volume.

A cone is essentially a pyramid with a circular base, and it follows the exact same pattern: V = (1/3) × base area × height, where the base area happens to be πr² for a circle.

Yes — the 1/3 × base area × height relationship holds for any pyramid, whether its apex sits directly above the base's center (a "right" pyramid) or is shifted to one side (an "oblique" pyramid), as long as the height is still measured as the perpendicular distance from the apex to the base plane.

Free Pyramid Volume Calculator