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Regular Tetrahedron: Volume, Area & Sphere Radii

Relate an equal-edge tetrahedron’s edge, height, volume, surface area and inscribed sphere measurements.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A regular tetrahedron with edge 6√2 has volume 72 and midsphere radius 3 in consistent units. Doubling every edge multiplies volume by 8 and surface area by 4.

V = a³/(6√2); surface area = √3 a²; height = a√(2/3); circumsphere R = a√6/4; insphere r = a√6/12; midsphere radius = a√2/4

Worked example

Enter these known values and leave the other values blank.

Equal edge length
8.485281374238570292810132345258188471418031252261688439060078428 m
Perpendicular tetrahedron height
6.928 m
Tetrahedron volume
72 m³
Total area of four faces
124.7 m²
Surface area divided by volume
1.732 1/ m
Insphere radius (tangent to faces)
1.732 m
Midsphere radius (tangent to edges)
3 m
Circumsphere radius (through vertices)
5.196 m

Assumptions and limitations

  • This is a regular tetrahedron: all six edges are equal and all four faces are equilateral triangles. It does not apply to an arbitrary tetrahedron with unequal edges.
  • All dimensions, area and volume are positive. The zero-edge limit is excluded because the surface-to-volume ratio would be undefined.
  • The surface-to-volume ratio has inverse-length units. A value stated per meter differs numerically from the same ratio stated per centimeter.
  • The insphere touches faces, the midsphere touches edges and the circumsphere passes through vertices. These are distinct radii.

Common questions

Can I start from volume instead of edge length?

Yes. Enter a positive volume in its selected unit to obtain the positive regular-tetrahedron edge and its other measurements.

Does this work for any triangular pyramid?

No. The formulas assume six equal edges. An irregular triangular pyramid needs additional dimensions or coordinates.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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