Tetrahedron Calculator
A regular tetrahedron is the simplest of the five Platonic solids: a three-dimensional shape with four faces, each an equilateral triangle of the same size. All six edges are equal length, all four vertices are equivalent, and it has the highest symmetry possible for a solid with triangular faces. You find it in crystal structures, molecular geometry (the methane molecule is tetrahedral), dice design, and architectural frameworks. Because of its high symmetry, every property of a regular tetrahedron follows from a single measurement: the edge length a. This calculator takes that edge length and returns five key properties. Volume tells you the interior space enclosed. Surface area is the combined area of the four equilateral triangular faces. Height is the perpendicular distance from a face to the opposite vertex. The circumradius is the radius of the sphere that passes through all four vertices, and the inradius is the radius of the sphere that fits perfectly inside touching all four faces. Notably, the circumradius is always exactly three times the inradius. Enter your edge length in any unit, millimetres, centimetres, metres, and the other values come back in the same unit or the appropriate square and cubic units. The calculator is designed for students studying geometry, engineers working with tetrahedral structures, and anyone who needs a fast reference for these classic solid geometry formulas.
All formulas apply to a regular tetrahedron only (all four faces are equilateral triangles of equal size). For irregular tetrahedra the formulas differ.
How it works
For a regular tetrahedron with edge length a, the five properties are:
Volume: V = a³ / (6√2)
Surface area: SA = √3 × a²
Height: h = a × √(2/3)
Circumradius: R = a × √6 / 4
Inradius: r = a / (2√6)
The circumradius is always three times the inradius (R = 3r). Each face has area (a²√3)/4 and the total surface area is four times this.
Worked example
With edge length a = 5: volume = 125 / (6√2) = 14.73 cubic units. Surface area = √3 × 25 = 43.30 square units. Height = 5 × √(2/3) = 4.08 units. Circumradius = 5 × √6 / 4 = 3.06 units. Inradius = 5 / (2√6) = 1.02 units. Note that 3.06 / 1.02 = 3, confirming the R = 3r relationship.
Related calculators
- Ellipsoid Volume Calculator: volume and surface area from three semi-axes.
- Capsule Volume Calculator: volume and surface area of a cylinder with hemispherical ends.
- SA:V Ratio Calculator: compare how surface area scales against volume for different shapes.
- Law of Sines and Cosines: solve triangle problems used in face calculations.