About the Tetrahedron Calculator
A tetrahedron is the simplest solid: four triangular faces, six edges and four corners. A regular tetrahedron — one of the five Platonic solids — has four identical equilateral triangles as faces. It appears as the four-sided die (d4), methane and other molecules with tetrahedral bonding, tetra-pak cartons, space-frame trusses and caltrops.
In regular mode, enter whichever measurement you know — the edge length, the volume, the total surface area or the height — and this tetrahedron calculator solves for the edge and returns everything else: volume, surface area, area of one face, height, inradius, midradius, circumradius and the dihedral angle between faces.
In vertex mode, paste the 3D coordinates of any four points (regular or not) and the calculator returns the volume from the scalar triple product, the total surface area and the six edge lengths. That is useful for meshes, surveying volumes and vector homework.
With the default inputs, the volume is 25.4558. Change any value above to recalculate instantly.
How to use the tetrahedron calculator
- 1Choose a regular tetrahedron or any tetrahedron from four vertices.
- 2For a regular one, pick what you know (edge, volume, area or height) and enter it.
- 3For vertices, type the x y z coordinates of the four corners, one per line.
- 4Read the volume, surface area and the other measurements.
Formula and method
A regular tetrahedron with edge a has four equilateral faces of area (√3/4)a², so its total surface area is √3·a². Its height (from a corner perpendicular to the opposite face) is a√(2/3), and the volume — one third of base area times height — simplifies to a³/(6√2) ≈ 0.11785a³. The inscribed sphere has radius a/√24, the circumscribed sphere a√(3/8) (exactly three times the inradius), and adjacent faces meet at arccos(1/3) ≈ 70.53°.
For any tetrahedron with corners A, B, C, D, the three edge vectors from A span a parallelepiped whose volume is the absolute scalar triple product; the tetrahedron fills exactly one sixth of it. Each face area is half the length of the cross product of two of its edge vectors.
- a
- Edge length of the regular tetrahedron
- V, A
- Volume and total surface area
- h
- Height from a vertex to the opposite face
- r, R
- Inradius and circumradius
- A, B, C, D
- Vertex coordinates (vertex mode)
Worked examples
Regular tetrahedron with edge 6
V = 6³/(6√2) = 216/8.485 ≈ 25.46 cubic units and the surface is √3 × 36 ≈ 62.35 square units. The height is 6 × 0.8165 ≈ 4.90, the inscribed sphere has radius ≈ 1.22 and the circumscribed sphere ≈ 3.67.
Regular tetrahedron holding 1,000 cm³
Solving a³ = 6√2 × 1,000 gives an edge of about 20.40 cm. Such a tetrahedron is about 16.65 cm tall and needs roughly 720.6 cm² of material for its four faces.
Corner tetrahedron from vertices (0,0,0), (4,0,0), (0,3,0), (0,0,5)
The three edges from the origin are perpendicular, so the triple product is 4 × 3 × 5 = 60 and the volume is 60/6 = 10. The three right-triangle faces have areas 6, 10 and 7.5, and the slanted face adds about 13.87, for a total surface of about 37.37.
Frequently asked questions
What is the volume of a regular tetrahedron?+
V = a³/(6√2) ≈ 0.11785 × a³, where a is the edge length. An edge of 6 gives 25.46 cubic units; an edge of 10 gives about 117.85 cubic units.
How do you find the volume of a tetrahedron from coordinates?+
Take the three edge vectors from one vertex, compute their scalar triple product (the 3×3 determinant) and divide its absolute value by 6. If the result is zero the four points lie in one plane.
What is the height of a regular tetrahedron?+
h = a × √(2/3) ≈ 0.8165 × a. It runs from one vertex to the centroid of the opposite face, so a tetrahedron with 10 cm edges stands about 8.16 cm tall.
How many faces, edges and vertices does a tetrahedron have?+
Every tetrahedron has 4 triangular faces, 6 edges and 4 vertices. It is the only convex polyhedron in which every vertex is joined to every other vertex by an edge.
What is the angle between faces of a regular tetrahedron?+
The dihedral angle is arccos(1/3) ≈ 70.53°. The angle between two lines from the centre to two vertices (the tetrahedral bond angle in methane) is arccos(−1/3) ≈ 109.47°.