Tetrahedron Volume Calculator — V = a³√2 / 12
Enter the edge length of a regular tetrahedron to instantly compute its volume, surface area, height, circumradius, inradius and midradius — with an animated 3-D view.
units
V = a³√2 / 12 (regular tetrahedron, all edges equal)
7.54 u³
- 1
Cube the edge length
a³ = 4³ = 64 - 2
Multiply by √2
64 × √2 = 90.509668 - 3
Divide by 12
90.509668 ÷ 12 = 7.5425
How does this calculator work?
A regular tetrahedron with edge a has volume V = a³√2/12, surface area √3·a² (four equilateral triangular faces), height a√6/3, circumradius R = a√6/4 and inradius r = a√6/12 = R/3. One edge length fully determines all properties. Applies only to the regular (all-equal-edge) case.
Formula
How this is calculated
A regular tetrahedron is the simplest Platonic solid: four equilateral triangular faces, four vertices, and six equal edges each of length a. Because all faces and edges are identical, one edge length fully determines every geometric property.
The volume V = a³√2/12 follows from dividing a cube of side s = a/√2 into exactly six congruent tetrahedra: cube volume = s³ = (a/√2)³ = a³/(2√2), divided by 6 = a³/(12√2) = a³√2/12. The surface area is four times the area of one equilateral triangle: SA = 4 × (√3/4)·a² = √3·a². The perpendicular height from a base to the opposite apex is h = a√6/3. The circumradius R = a√6/4 is the radius of the sphere passing through all four vertices; the inradius r = a√6/12 = R/3 is the radius of the inscribed sphere tangent to all four faces; and the midradius ρ = a√2/4 touches all six edges.
These formulas apply only to a regular tetrahedron. An irregular tetrahedron (with differing edge lengths) cannot be computed from a single length — you would need all four face areas or all six edge lengths.
Frequently asked questions
A regular tetrahedron is a three-dimensional solid with four equilateral triangular faces, four vertices and six equal-length edges. It is one of the five Platonic solids — the one with the fewest faces — and is self-dual (a tetrahedron maps to a tetrahedron under face-to-vertex duality).
One derivation: a cube of side s can be divided into exactly six congruent regular tetrahedra each with edge a = s√2. The cube volume is s³ = (a/√2)³ = a³/(2√2). Dividing by 6 gives a³/(12√2) = a³√2/12. Alternatively, integrating triangular cross-sections from apex to base yields the same result.
For a regular tetrahedron R = 3r — the circumradius (sphere through all four vertices) is exactly three times the inradius (sphere tangent to all four faces). This 3:1 ratio is a distinctive geometric property. Equivalently, the centre of gravity divides the median in ratio 3:1 from vertex to face.
Also known as
TG we-Calculate Editorial Team. (2026). Tetrahedron Volume Calculator — V = a³√2 / 12 [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/tetrahedron-volume-calculator
TG we-Calculate Editorial Team. "Tetrahedron Volume Calculator — V = a³√2 / 12." TG we-Calculate. 2026. https://we-calculate.com/calculator/tetrahedron-volume-calculator.
TG we-Calculate Editorial Team, "Tetrahedron Volume Calculator — V = a³√2 / 12," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/tetrahedron-volume-calculator
@misc{wecalculate_tetrahedron_volume_calculator, title = {Tetrahedron Volume Calculator — V = a³√2 / 12}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/tetrahedron-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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