01What this calculator tells you
This calculator finds the volume of a triangular pyramid — a solid with a triangular base and three triangular sides meeting at an apex. It uses the universal pyramid rule, V = ⅓ × base area × height, and works out the base area for you whether you know the base’s base and height, its three sides, or the area outright. There is also a one-tap mode for a regular tetrahedron from a single edge length.
Enter your figures in any unit and you get the volume in the matching cubic unit, the intermediate base area, an alternate cubic unit, and litres where it makes sense. The one-third factor is the same one behind our area tools; browse the full set on our calculators home page.
02Which inputs you need
The only quantity you always need is the pyramid’s perpendicular height H — the straight-line distance from the base up to the apex. How you supply the base area depends on what you can measure, as summarised below.
03Why the volume is one-third of the prism
A triangular pyramid on a given base holds exactly one-third of the triangular prism built on the same base and height — the same factor that relates a cone to its cylinder. That is why the formula has a ⅓ in it rather than being simply base × height. This is a classic result of solid geometry, proved in Euclid’s Elements Book XII (Clark University) and revisited with calculus in these Lamar University volume notes.
For a regular tetrahedron — four identical equilateral triangles — the height is fixed by the edge length, so the whole volume collapses to a single formula, V = a³√2 ÷ 12 ≈ 0.1179 a³. If you need the base area on its own, our square meter calculator handles triangle areas in real-world units.
- Decide how you know the base: base & height, three sides, or a known area — then pick that mode.
- Choose your measurement unit; the volume returns in the matching cubic unit.
- Enter the base figures, then the pyramid’s perpendicular height H (skip H for a regular tetrahedron).
- Press Calculate to see the base area and the volume, with cubic-unit conversions.
To measure H on a physical model, stand the pyramid on its base and measure straight up to the apex with a set square — do not follow a sloping edge. If you only have the slant edge, the Pythagorean theorem (Lamar University) converts it to the true height. Ready? Enter your values and hit Calculate.
- Use the perpendicular height. H is the vertical distance to the apex, not the length of a slanted edge or face.
- Base height ≠ pyramid height. The triangle’s own height h feeds the base area; the pyramid height H is separate.
- Three sides must form a triangle. Each side must be shorter than the sum of the other two, or no triangle (and no volume) exists.
- The tetrahedron mode assumes a regular tetrahedron. All six edges equal; an irregular one needs the general base-and-height mode.
- Keep units consistent. Every length must use the same unit before the volume is meaningful.
04Related calculators
Working through a related project? Try our Stud Calculator, Board and Batten Calculator, and Barndominium Material Calculator.
01The formula
Every triangular pyramid follows the same rule: one-third of the base area times the perpendicular height. The only variation is how you get the base area of the triangle.
Where:
- V= the volume of the pyramid, in cubic units.
- B= the area of the triangular base, in square units.
- H= the perpendicular height of the pyramid, base to apex.
- b, h= the base triangle’s own base and perpendicular height.
- a, b, c= the three side lengths of the base triangle.
02Worked example
Take a pyramid whose base is a 3-4-5 right triangle and whose height is 10 cm. Using the three-sides route, work it one line at a time:
So the pyramid holds 20 cm³. Notice Heron’s formula and the ½bh shortcut give the same base area of 6 cm² — a handy cross-check whenever the base is a right triangle.