Tetrahedron in 3D
The simplest solid.
Four triangles enclose a space. This regular tetrahedron has four identical equilateral faces and six equal edges.
On this page6 sections
Interactive model loading. Reference images and example calculations are available below.
A regular tetrahedron with four equilateral triangular faces
What is a tetrahedron?
A tetrahedron is a polyhedron with four triangular faces. The regular tetrahedron shown here has equilateral faces and equal edges. It is one of the five Platonic solids, and it is also a triangular pyramid.
One base. Three sides. Four equal triangles.
01 / Build it from four faces
Fold a triangle
around each side.
The three outer triangles hinge upward until their free edges meet. Nothing stretches: every face remains an equilateral triangle throughout the fold.
The angle between two faces is different: about 70.53°.
02 / Look inside
Tetrahedron height:
h = √(2/3)a.
The height drops straight from the top vertex to the center of the opposite face. It meets that face at a right angle.
For edge length a, the height is √(2/3)a, about 81.65% of the edge. That gives the pyramid formula V = ⅓ × base area × height.
Why the height is shorter than an edge.
The foot of the height lies at the center of the equilateral base. That center is a / √3 from any base vertex. The edge a is the hypotenuse of a right triangle, so:
h² + (a / √3)² = a²
h = √(2/3)a ≈ 0.816497a
At edge a = 4, the height is units. This perpendicular height belongs in V = ⅓ × base area × height.
03 / Measurements
From edge a = 4.
All results follow the size control above.
| Measurement | Formula | Result |
|---|---|---|
| Volume | a³ / (6√2) | units³ |
| Surface area | √3a² | units² |
| Area of one face | √3a² / 4 | units² |
| Height | √(2/3)a | units |
| Center to a face | √6a / 12 | units |
| Center to a vertex | √6a / 4 | units |
Values are rounded to three decimal places. Formulas apply to a regular tetrahedron.
The solid, taken apart.
A few useful distinctions.
Is every tetrahedron regular?
No. Four triangular faces define a tetrahedron. A regular tetrahedron adds equal edge lengths and four equilateral faces.
Is a tetrahedron a pyramid?
Yes. Any one of its triangular faces can be treated as the base, with the opposite vertex as the apex.
What changes if I double the edge?
The surface area becomes four times as large and the volume becomes eight times as large. The face, edge and vertex counts stay the same.
From an edge
to the height.
The complete tetrahedron keeps its perpendicular height beside the volume formula, with a folding net and separated faces below.
- h = √(2/3) · a
- Height runs from a vertex to the center of the opposite face.
- V = a³ / (6√2)
- One edge determines the volume of the regular solid.
- About 70.53°
- The interior dihedral angle lies between adjacent faces.