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Tetrahedron Surface Area Calculator

Enter the edge length of the regular tetrahedron.

How it is calculated

The tool is for a regular tetrahedron: four identical equilateral triangles, so one edge length a fixes everything. It returns four values:

face = (√3 / 4) · a²
surface area = √3 · a² (four faces)
volume = a³ / (6√2)
height = a · √(2/3) (vertex to the opposite face)

The edge must be above zero. Results are rounded to four decimals and labelled with generic units: u for length, u² for area and u³ for volume. They are in whatever unit you used for the edge, so centimetres in give cm², cm³ and cm out.

Worked example

Take an edge of a = 5. One face is (1.7321 / 4) × 25 = 10.8253. Four faces give a surface area of √3 × 25 = 43.3013 u². The volume is 125 / (6 × 1.4142) = 14.7314 u³, and the height is 5 × 0.8165 = 4.0825 u. If the edge was 5 cm, the surface area is 43.3013 cm².

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Common mistakes

  • Using it for an irregular tetrahedron such as a triangular pyramid with a right-angle corner; the formulas need all six edges equal.
  • Entering the height of the solid or the slant height of a face instead of the edge length.
  • Quoting the area of one face as the surface area; the surface area is four times the face.
  • Forgetting the units are squared and cubed: doubling the edge multiplies the area by 4 and the volume by 8.
  • Mixing units, for example an edge in millimetres but expecting an answer in square metres.

About this calculator

A regular tetrahedron is a Platonic solid made of four equilateral triangular faces. Its total surface area follows directly from the area of one equilateral triangle: with edge length a, a single face has area (√3⁄4)a², so the total surface area is A = √3 × a². Its volume is V = a³ ÷ (6√2), and its height (from one vertex perpendicular to the opposite face) is h = a√(2/3).

Because all four faces, edges, and vertex angles are identical, a single edge length fully determines every other measurement of the solid. Geometry students, 3D designers, and engineers working with tetrahedral trusses or molecular models (tetrahedral geometry shows up naturally in methane's molecular structure and in space-frame construction) use these formulas to get surface area, volume, and height from one known edge length.

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Comments (1)

  • Editorial team

    If you are building a physical model, the surface area figure is the amount of sheet material in the four triangles alone, with no seams, tabs or waste. An edge of 10 cm gives 173.2051 cm², and the net is simply four equilateral triangles of 10 cm per side joined along their edges. Add an allowance for glue flaps and cutting. Are you folding paper, cutting cardboard or modelling this in 3D software?

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