How this instrument works
A triangular pyramid, also called a tetrahedron, is a solid with a triangular base and three more triangular faces meeting at a single apex point above it. Its volume follows the general pyramid rule shared by every pyramid regardless of the base's shape: volume equals one third of the base area multiplied by the height (the perpendicular distance from the apex down to the base's plane). For a triangular base this becomes V = (base area × height) ⁄ 3.
This calculator takes the base's AREA directly rather than assuming a perfectly regular tetrahedron with every edge equal — a genuinely different, more general tool. A regular tetrahedron's volume can be computed from a single edge length alone, since every dimension is locked to that one number, but a general triangular pyramid can lean, skew, or have a scalene base of any shape, and the base-area-and-height version handles all of those cases as long as that area and height are already known or computed separately.
The one-third factor is not arbitrary — it comes from a genuine geometric fact: three tetrahedra of equal base and height can be assembled to exactly fill a triangular prism of that same base and height, with no gaps and no overlap. That decomposition, provable directly with Cartesian coordinates and integration, is the reason a cone also holds one third of its matching cylinder's volume, and a general pyramid one third of its matching prism's — the same one-third relationship, independent of the base's shape.
- Compute or enter the triangular base's area into the Base area field.
- Enter the pyramid's height — the perpendicular distance from the apex straight down to the base's plane — into the Height field.
- Read Volume: the sheet multiplies base area by height and divides by 3.
- For a REGULAR tetrahedron (every edge equal), the Tetrahedron Volume Calculator elsewhere on this site computes the same result directly from a single edge length.
Worked example — a base area of 6 with a 9-unit height
A triangular pyramid has a base of area 6 square units, and its apex sits 9 units above the base's plane, measured perpendicularly. The volume is V = 6 × 9 ⁄ 3 = 18 cubic units — exactly one third of the 54 cubic units a matching triangular prism (same base, same height) would hold.
Compare a pyramid with a larger base area of 12 and a shorter height of 5: V = 12 × 5 ⁄ 3 = 20 cubic units — a comparable total volume reached through a wider, flatter shape instead of a narrower, taller one, since the formula only cares about the product of the two, not their individual proportions.
Questions
What is the formula for the volume of a triangular pyramid?
V = (base area × height) ⁄ 3, where base area is the area of the triangular base and height is the perpendicular distance from the apex to the base's plane. This is the general pyramid volume rule, valid for any base shape, applied specifically to a triangular base.
Why is there a division by 3 in the formula?
Because exactly three pyramids of equal base and height can be assembled, with no gaps or overlaps, to fill a prism sharing that same base and height. That geometric fact, provable through coordinate integration, is why every pyramid — triangular, square, or otherwise — holds exactly one third of its matching prism's volume.
How is this different from the Tetrahedron Volume Calculator?
The tetrahedron page assumes a REGULAR tetrahedron, where all four faces are identical equilateral triangles, and computes volume from a single edge length. This page is more general: it accepts any triangular base's area and any height, correctly handling irregular, leaning, or scalene tetrahedra the single-edge-length formula cannot.
What if the base area is zero?
The volume is zero regardless of the height — a degenerate pyramid with no base to speak of, collapsed to a line or a single point.
Does the apex need to sit directly above the base's center?
No — the height is defined as the perpendicular distance from the apex to the base's PLANE, not to any particular point within the base. An apex positioned anywhere above that plane, even far to one side, gives the same volume as long as the perpendicular height and the base area stay the same.