Tetrahedron Calculator: Volume, Surface Area, Height and Radii
Regular tetrahedron from any one value (edge, height, volume, area or radius), or any tetrahedron from its 4 vertices: volume, surface area, height, in/circumradius.
How it works
Regular, edge a: V = a³ ÷ (6√2) = (√2 ÷ 12)a³ · S = √3 a² · h = a√(2/3) · R = a√6 ÷ 4 · r = a√6 ÷ 12 · midradius = a√2 ÷ 4 · dihedral angle = arccos(1/3) ≈ 70.53° · Any tetrahedron with vertices A, B, C, D: V = |AB · (AC × AD)| ÷ 6 · face area = ½|cross product| · r = 3V ÷ S
A regular tetrahedron is the triangular pyramid whose four faces are identical equilateral triangles. One length fixes everything, so you can enter whichever you have: the edge, the height, the volume, the surface area, the circumradius or the inradius. The calculator converts it back to the edge and then applies the standard formulas, giving the volume, surface area, height, the area and slant height of one face, the radii of the circumscribed, inscribed and mid spheres, the surface-to-volume ratio and the two characteristic angles (70.53° between faces, 54.74° between an edge and a face). For a tetrahedron that is not regular, switch to “Any (4 vertices)” and type the coordinates of its corners. The volume is one sixth of the absolute value of the scalar triple product of the three edge vectors from one vertex, each face area is half the length of a cross product, the inradius is 3V ÷ S, and the circumcenter comes from solving three linear equations. The pyramid volume calculator needs a base area and a height; this page works from a single measurement of a regular tetrahedron or from raw coordinates of any tetrahedron.
Worked example
A regular tetrahedron with 10 cm edges: V = 10³ ÷ (6√2) = 1,000 ÷ 8.4853 = 117.8511 cm³; S = √3 × 100 = 173.2051 cm²; h = 10 × √(2/3) = 8.165 cm; each face has area 43.3013 cm². The circumscribed sphere has radius 10√6 ÷ 4 = 6.1237 cm, the inscribed sphere 10√6 ÷ 12 = 2.0412 cm (and 6.1237 + 2.0412 = 8.165 = h), and the faces meet at 70.5288°.
Sources: Wolfram MathWorld: Regular Tetrahedron · Omni Calculator: Tetrahedron volume (80 cm worked example) · Rechneronline: Tetrahedron calculator
Frequently asked questions
What is the volume of a regular tetrahedron?
V = a³ ÷ (6√2), about 0.11785 a³. A tetrahedron with 80 cm edges holds 80³ ÷ 8.4853 ≈ 60,340 cm³, or about 0.0603 m³.
How tall is a regular tetrahedron?
h = a × √(2/3) ≈ 0.8165 a, measured from a vertex straight down to the opposite face. A 10 cm tetrahedron is 8.165 cm tall.
How do I find the edge from the volume?
Reverse the volume formula: a = ∛(6√2 × V). A volume of 117.85 cm³ gives a = ∛1,000 = 10 cm. Choose “Volume” under “What do you know?” to do this automatically.
How do I find the volume of a tetrahedron from its vertices?
Take the three edge vectors from one vertex, AB, AC and AD, compute the scalar triple product AB · (AC × AD) and divide its absolute value by 6. For A(0,0,0), B(4,0,0), C(0,3,0), D(0,0,2) the product is 24, so V = 4.
What is the angle between two faces of a tetrahedron?
In a regular tetrahedron the dihedral angle is arccos(1/3) ≈ 70.53°. The angle between an edge and a face it meets is arccos(1/√3) ≈ 54.74°.
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This calculator is for educational purposes. Double-check important results before acting on them.