Tetrahedron Volume Calculator
Type the edge length of a regular tetrahedron — the calculator returns volume, face area, total surface area, height, insphere and circumsphere radii using exact square roots.
Volume (V = a³ / (6·√2))
25.455844 cubic units
- Face area (√3/4 · a²)
- 15.59
- Total surface area (√3 · a²)
- 62.35
- Height (h = a·√(2/3))
- 4.9
- Insphere radius (a / (2·√6))
- 1.22
- Circumsphere radius (a·√6 / 4)
- 3.67
A regular tetrahedron is the simplest Platonic solid: four equilateral-triangle faces, six equal edges, four vertices. A single edge length a fixes every measurement. Volume V = a³/(6·√2) ≈ 0.1178·a³. Surface area is four triangle faces: 4·(√3/4)·a² = √3·a². The height from base to apex is h = a·√(2/3). The largest sphere that fits inside touches all four faces at their centres, with radius a/(2·√6); the smallest sphere that contains it passes through all four vertices, with radius a·√6/4.
How to use this calculator
Type the edge length a of the regular tetrahedron in any linear unit you like — centimetres, metres, inches or feet. All six edges of a regular tetrahedron are equal, so a single edge length is all the geometry needs. The calculator returns the volume in cubic units of that unit, alongside the area of one triangular face, the total surface area of all four faces, the height from base to apex, the insphere radius (the largest sphere that fits inside) and the circumsphere radius (the smallest sphere that contains it). Linear outputs use the same unit as a, areas use squared units, the volume uses cubed units.
How the calculation works
A regular tetrahedron is the simplest Platonic solid: four equilateral-triangle faces, six equal edges of length a, four vertices, all completely symmetric. Because every edge is the same length, a single number defines the entire solid. Volume comes from integrating the triangular cross-section along the height: V = a³/(6·√2), roughly 11.78% of the cube of the edge. Total surface area is four equilateral triangles of side a, and each has area (√3/4)·a², giving √3·a² together. The height from any face to the opposite vertex is h = a·√(2/3). The largest sphere that fits inside touches all four faces at their centres, with insphere radius r_in = a/(2·√6); the smallest sphere that contains the whole solid passes through all four vertices, with circumsphere radius r_out = a·√6/4. Square roots are kept at full machine precision (~16 significant figures); rounding only happens at display.
Worked example
Take a regular tetrahedron with edge length a = 6. Volume V = 6³/(6·√2) = 216/(6·√2) = 36/√2 = 18·√2 ≈ 25.455844 cubic units. One face area = (√3/4)·36 = 9·√3 ≈ 15.588457 square units. Total surface area = √3·36 = 36·√3 ≈ 62.353829 square units, exactly four times a single face. Height h = 6·√(2/3) = 2·√6 ≈ 4.898979 units. Insphere radius = 6/(2·√6) = √6/2 ≈ 1.224745. Circumsphere radius = 6·√6/4 = 3·√6/2 ≈ 3.674235. Sanity check: the circumsphere radius is exactly three times the insphere radius, a defining property of the regular tetrahedron.
Frequently asked questions
What is the formula for the volume of a regular tetrahedron?
V = a³ / (6·√2), where a is the edge length. Equivalently, V = a³·√2/12 ≈ 0.11785·a³. All six edges of a regular tetrahedron have the same length, so a single edge fully determines the volume — no separate height or base area is needed.
What is a tetrahedron?
A tetrahedron is any polyhedron with four triangular faces, four vertices, and six edges — the simplest solid in three dimensions. A regular tetrahedron is the special case where all four faces are congruent equilateral triangles and all six edges are equal. It is one of the five Platonic solids and is completely defined by a single edge length.
How is the height of a regular tetrahedron related to its edge?
The perpendicular height h from any triangular face to the opposite vertex is h = a·√(2/3) = a·√6/3 ≈ 0.8165·a. Combined with the base area of an equilateral triangle, (√3/4)·a², this gives V = (1/3)·base·height = (1/3)·(√3/4)·a²·a·√(2/3) = a³/(6·√2), matching the compact formula.
Does this work for an irregular tetrahedron?
No — every formula here assumes a regular tetrahedron, where all six edges are equal. An irregular (general) tetrahedron is defined by four vertex positions and needs the scalar triple product V = |(b−a)·((c−a)×(d−a))|/6, which is not captured by a single edge length. If your tetrahedron has unequal edges, split it into base area and height and use the pyramid volume formula V = (1/3)·base·height instead.
How do I convert the volume to litres or gallons?
If your edge length is in centimetres, the volume comes out in cubic centimetres, and 1 cm³ = 1 millilitre, so divide by 1000 to get litres. In metres, the volume is in m³ and 1 m³ = 1000 litres. For US gallons, 1 US gallon ≈ 3.78541 litres; for UK gallons, ≈ 4.54609 litres. Use the Volume Converter for any other unit pair.
What is the relationship between the insphere and circumsphere?
The circumsphere radius of a regular tetrahedron is exactly three times its insphere radius: r_out / r_in = 3. Numerically r_in = a/(2·√6) ≈ 0.2041·a and r_out = a·√6/4 ≈ 0.6124·a. This 3:1 ratio is unique to the regular tetrahedron among the Platonic solids and follows from the geometry that its centroid divides the height in a 1:3 ratio between the face and the opposite vertex.