How this calculator works
A regular tetrahedron is the simplest Platonic solid: four equilateral triangles forming a pyramid with a triangular base. One edge length fixes everything else. The volume works out to a³ divided by six times the square root of two, which is about 0.11785 a³. The height is the distance from a base corner to the opposite face, not to a corner, and the insphere and circumsphere radii are in a fixed 1 to 3 ratio. The tetrahedron also satisfies Euler's formula with 4 faces, 6 edges and 4 vertices.
Formula
V = a³/(6√2); surface area = √3 a²; height = a√(2/3); circumradius = a√6/4; inradius = a√6/12.
Worked example
An edge of 3 m gives a volume of about 3.182 m³, a surface area of about 15.588 m² and a height of about 2.449 m. The circumsphere radius is about 1.837 m and the insphere radius is about 0.612 m.
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Frequently asked questions
What makes a tetrahedron regular?
A regular tetrahedron has four identical equilateral triangle faces, so all six edges have the same length. An irregular tetrahedron can still be solved from other data, but this calculator is for the regular case.
What is the difference between the height and a slant height?
The height runs from a vertex perpendicular to the opposite face. Each face has its own slant height measured within that triangular face, which is shorter than the full height.
Why is the circumradius three times the inradius?
In a regular tetrahedron the centre of the insphere and the circumsphere coincide. That centre divides the height in a 1 to 3 ratio, which fixes the two radii at a√6/12 and a√6/4.