What is a regular tetrahedron?
A regular tetrahedron is the simplest Platonic solid: four congruent equilateral-triangle faces joined so that every edge has the same length. It has 4 faces, 6 edges and 4 vertices. Because every regular tetrahedron is determined by one edge length a, volume, surface area, height and sphere radii all scale from that single dimension.
It is also a triangular pyramid. The important distinction is the word regular: all four faces are equilateral triangles, not four arbitrary triangles. That symmetry is what makes the formulas below so compact.
Tetrahedron formulas from one edge
Let a be the common edge length. Once a is known, every other standard measurement follows directly.
Volume
- V
- volume enclosed by the four faces
- a
- common edge length
Total surface area
- S
- area of all four equilateral faces
- a
- common edge length
Height
- h
- perpendicular distance from a vertex to the opposite face
- a
- common edge length
Face area and useful radii
Scaling rule: if the edge doubles, height doubles, surface area becomes 4 times as large, and volume becomes 8 times as large. This is why the live scaling chart is useful: it shows the difference between linear, square and cubic growth.
Why h = a√6/3
Drop a perpendicular from the top vertex to the opposite triangular face. The foot lands at the center of the equilateral base. The distance from the center of an equilateral triangle to any vertex is a√3/3. Applying the Pythagorean theorem to the resulting right triangle gives the tetrahedron height.
Tetrahedron calculation examples
The examples below use the same formulas as the calculator. They are deliberately concrete so a student can reproduce the arithmetic by hand.
a = 5 cm
For a 5 cm regular tetrahedron:
S = √3·25 ≈ 43.301 cm² and h = 5√6/3 ≈ 4.082 cm.
V = 100 cm³
Work backward with the cube root:
The calculator then derives surface area, height and sphere radii from that edge.
S = 100 cm²
Rearrange S = √3a²:
This inverse form is useful when material coverage is known before the physical edge is chosen.
Double the edge
If a changes from 4 cm to 8 cm:
This cubic scaling is important in manufacturing, storage and mesh-size calculations.
Where tetrahedra are used
A tetrahedron is more than a textbook solid. Its regular geometry appears in physical objects, molecular models, structural systems and numerical simulation.
Four-sided dice
A regular tetrahedron is the familiar geometric form of a d4. Its four triangular faces correspond to the four possible face outcomes.
Molecular geometry
Methane has four hydrogen atoms arranged tetrahedrally around carbon. The H–C–H bond angle is about 109.47°, a different angle from the 70.53° dihedral angle between faces.
Engineering & simulation
Tetrahedral elements are widely used to divide complex three-dimensional regions into smaller cells for finite-element and deformation simulations.
Faces, edges, vertices and spheres
The four faces are congruent equilateral triangles. Each face has area (√3/4)a², so four faces give S = √3a². The symmetry also creates three natural spheres: the insphere touches every face, the midsphere touches every edge, and the circumsphere passes through all four vertices.
| Property | Relationship | Meaning |
|---|---|---|
| Inradius rᵢ | a/(2√6) | Sphere tangent to all four faces |
| Midsphere rₘ | a/√8 | Sphere tangent to all six edges |
| Circumradius R | a√6/4 | Sphere through all four vertices |
| Surface / volume | 6√6/a | Surface-to-volume ratio |
| Dihedral angle | arccos(1/3) | ≈ 70.5288° between adjacent faces |
How to read the result
Volume measures capacity
Volume tells you how much three-dimensional space is enclosed. It uses cubic units such as cm³, m³ or in³.
Surface area measures exposure
Surface area tells you how much exterior material covers the four faces. It uses square units.
Height is linear
The vertex-to-face height is 0.8165 times the edge. It is not the altitude of one triangular face.
Radii describe symmetry
Inradius, midsphere radius and circumradius locate natural spheres created by the tetrahedron's symmetry.
Tetrahedron calculator FAQ
It is a Platonic solid made from four congruent equilateral triangles. It has 4 faces, 6 equal edges and 4 vertices.
For a regular tetrahedron with edge a, use V = a³/(6√2), equivalently V = (√2/12)a³.
Use h = a√6/3. The height is the perpendicular distance from a vertex to the opposite triangular face.
The total surface area is S = √3a² because each of the four equilateral faces has area (√3/4)a².
The regular tetrahedron's dihedral angle between adjacent faces is about 70.53°. In methane, the H–C–H bond angle between two center-to-vertex directions is about 109.47°. They describe different geometric relationships.
Yes. Rearrange the volume formula to a = ∛(6√2V). The calculator supports edge, volume, surface area and height as the known quantity.
Height doubles, surface area becomes four times larger and volume becomes eight times larger. This follows from linear, square and cubic scaling.
No. The Great Pyramid is a square-based pyramid with five faces. A regular tetrahedron has four triangular faces and no square base.