Triangular Pyramid Volume Calculator
A triangular base with a base of 6 and height of 4, topped by a pyramid height of 9, has a volume of 36 cubic units. This triangular pyramid volume calculator finds the volume of a tetrahedron, a pyramid whose base is a triangle instead of a square or rectangle. It works in two modes: build the base from a triangle base and height plus the pyramid's perpendicular height, or enter a single edge length for a regular tetrahedron, where all four faces are congruent equilateral triangles. This is a different shape than the rectangular-base pyramid-volume-calculator on this site, which assumes a four-sided base and needs a base length and width. A triangular pyramid only has three base edges, so the base area comes from a triangle formula instead of a rectangle formula, and a regular tetrahedron collapses all four faces to a single edge length.
Quick answer
A triangular pyramid, or tetrahedron, has a triangular base and three triangular sides meeting at one apex.
Volume
36 cubic units
Base Area
12
What this tells you
- •A triangular pyramid, or tetrahedron, has a triangular base and three triangular sides meeting at one apex.
- •In base-height mode, the base area comes from 0.5 x triangle base x triangle height, then the volume is one-third of base area times the pyramid's perpendicular height.
- •In regular-tetrahedron mode, every face is an identical equilateral triangle, so one edge length alone determines the entire solid.
- •The one-third factor in the volume formula applies to every pyramid and cone, not just triangular ones, because a pyramid is one-third of the prism that shares its base and height.
- •Pyramid height means the perpendicular distance from the apex straight down to the base plane, not the slant height along an edge.
How to Use
- 1Choose base-height mode if you know the triangular base's own base and height, plus the pyramid's perpendicular height from the apex to the base plane.
- 2Enter the triangle base, triangle height, and pyramid height. The calculator finds the base area for you before applying the volume formula.
- 3Choose regular-tetrahedron mode if all four faces are identical equilateral triangles and you only know one edge length.
- 4Enter that single edge length and the calculator returns the volume along with the area of one equilateral face.
- 5Calculate to see the volume and the base area used, so you can check the intermediate step before trusting the final number.
How It Works
Formula
Base-height: Volume = (1/3) x (0.5 x triangleBase x triangleHeight) x pyramidHeight | Regular tetrahedron: Volume = edgeLength^3 / (6 x sqrt(2))In base-height mode, the calculator first finds the triangular base's area with the standard triangle formula, one-half base times height. It then applies the universal pyramid rule, Volume = (1/3) x base area x perpendicular height, where the height is measured straight up from the base plane to the apex, not along a slanted edge. For a base of 6, height of 4, and pyramid height of 9, the base area is 0.5 x 6 x 4 = 12, and the volume is (1/3) x 12 x 9 = 36. In regular-tetrahedron mode, all six edges are equal and all four faces are congruent equilateral triangles, which lets the whole volume collapse to one formula built from a single edge length: edgeLength cubed divided by 6 times the square root of 2. For an edge length of 5, that is 125 divided by 8.485281, or about 14.7314 cubic units.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Base-height: base 6, height 4, pyramid height 9
The triangular base area is 0.5 x 6 x 4 = 12. Multiplying by the pyramid height of 9 and dividing by 3 gives 36 cubic units.
Base-height: base 8, height 5, pyramid height 6
Base area is 0.5 x 8 x 5 = 20. Volume is (1/3) x 20 x 6 = 40 cubic units.
Regular tetrahedron: edge length 5
125 divided by 6 x sqrt(2), which is about 8.485281, gives approximately 14.7314 cubic units for a regular tetrahedron with every edge equal to 5.
Regular tetrahedron: edge length 1 (unit tetrahedron)
1 cubed divided by 6 x sqrt(2) gives about 0.117851 cubic units, the volume of the smallest regular tetrahedron with edge length 1.
Regular Tetrahedron Volume by Edge Length
Volume for a regular tetrahedron (all four faces equilateral) at common edge lengths.
| Edge Length | Volume |
|---|---|
| 1 | 0.1179 |
| 2 | 0.9428 |
| 5 | 14.7314 |
| 10 | 117.8511 |
Common mistakes
- Using the slant height along an edge instead of the perpendicular pyramid height. The volume formula needs the straight vertical distance from the apex to the base plane.
- Forgetting the one-third factor and calculating a prism volume instead of a pyramid volume. A triangular pyramid is exactly one-third of the triangular prism sharing its base and height.
- Trying to use the regular-tetrahedron formula on a triangular pyramid whose faces are not all congruent equilateral triangles. That single-edge shortcut only works when the solid is fully regular.
- Mixing up triangle base and triangle height with the pyramid's own base and height. The base-height mode has three separate inputs because the triangular base needs two of them just to find its area.
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