What this calculator does
A triangular pyramid, more formally a tetrahedron, has a triangular base and three more triangular faces meeting at a single apex above it. That is a genuinely different solid from the more familiar pyramid with a square or rectangular base: the base itself needs three side lengths rather than two, before the height even comes into it.
This calculator works out the volume of a triangular pyramid from its three base sides and its height, the perpendicular distance from the apex straight down to the base plane. The base area is found first with Heron’s formula, then combined with the height using the same one-third rule that applies to every pyramid and cone.
The formula
The base area comes from Heron’s formula: with semiperimeter s = (a+b+c)/2, base area = √(s(s−a)(s−b)(s−c)). Volume then follows the general pyramid rule, V = ⅓ × base area × height, where height is measured perpendicular to the base plane, not along a sloped edge.
| Term | Meaning |
|---|---|
| a, b, c | The three side lengths of the triangular base. |
| Height | The perpendicular distance from the apex straight down to the base plane, not the length of a sloped edge. |
| Regular tetrahedron | The special case where all three base sides and all three lateral edges are equal, so every face is the same equilateral triangle. |
The inputs explained
| Field | What to enter |
|---|---|
| Base side a | One side of the triangular base. |
| Base side b | A second side of the triangular base. |
| Base side c | The third side of the triangular base. |
| Height (apex above the base plane) | The perpendicular height from the apex down to the base plane. |
When to use it
A tetrahedral roof, tent or display stand
A tetrahedral tent, awning or point-of-sale display has a triangular pyramid volume that fixes how much air, or packed material, it actually encloses, worked out from the base sides and the apex height alone.
A geology or engineering wedge of material
A wedge-shaped rock volume, spoil heap corner or triangular fill section is often modelled as a triangular pyramid, and the volume here gives a reasonable estimate from three base measurements and a height.
Checking a volume of triangular pyramid homework answer
A geometry problem giving three base sides and a height is solved directly here, without separately applying Heron’s formula before 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.
How does volume change with height, at a fixed base?
An equilateral base with all three sides at 6, across a range of heights from short to tall.
| Height | Volume | Base area (triangular base) |
|---|---|---|
| 2 | 10.392 | 15.588 |
| 4 | 20.785 | 15.588 |
| 6 | 31.177 | 15.588 |
| 8 | 41.569 | 15.588 |
| 10 | 51.962 | 15.588 |
Volume of a regular tetrahedron across a range of edge lengths
A regular tetrahedron, where the height that makes every face an identical equilateral triangle is edge × √(2/3), across a range of edge lengths.
| Edge length (a = b = c, h set to match) | Volume | Base area (triangular base) |
|---|---|---|
| 2 | 15.776 | 5.916 |
| 4 | 30.170 | 11.314 |
| 6 | 41.569 | 15.588 |
| 8 | 47.703 | 17.889 |
| 10 | 44.222 | 16.583 |
Questions
What is a triangular pyramid?
A triangular pyramid, or tetrahedron, is a solid with a triangular base and three more triangular faces that meet at a single apex point above it. Every face is a triangle, unlike a square-based pyramid.
What is the formula for the volume of a triangular pyramid?
Volume = ⅓ × base area × height, the same one-third rule as any pyramid or cone. The base area itself comes from Heron’s formula applied to the three base side lengths.
Is the height the same as the length of a sloped edge?
No. Height is the perpendicular, straight-up distance from the apex to the base plane. A sloped lateral edge is longer than this perpendicular height and would overstate the volume if used in its place.
What makes a tetrahedron "regular"?
A regular tetrahedron has all four faces as identical equilateral triangles, which means all three base sides and all three lateral edges are the same length. Its height works out to exactly the edge length multiplied by √(2/3).
For a pyramid with a rectangular or square base instead, see the pyramid calculator. For the surface area and volume of a triangular prism, the shape formed by stretching a triangle along its length rather than tapering it to a point, see the surface area of a triangular prism calculator.