Calculate the volume, surface area, height, circumradius, inradius, and midsphere radius of regular and irregular tetrahedra from edge lengths.
A tetrahedron is the simplest three-dimensional polyhedron, consisting of four triangular faces, six edges, and four vertices. When all six edges are equal, it is called a regular tetrahedron — one of the five Platonic solids. The regular tetrahedron has the highest symmetry among triangular pyramids: every face is an equilateral triangle, every edge is the same length, and it can be inscribed in a sphere (circumsphere), with an insphere touching all four faces from inside.
The volume of a regular tetrahedron with edge length a is V = a³ / (6√2), and its surface area is √3 × a². The height — the perpendicular distance from any vertex to the opposite face — is a × √(2/3). The circumradius (R = a√6 / 4) and inradius (r = a√6 / 12) satisfy R = 3r, a distinctive property of regular tetrahedra.
For irregular tetrahedra, where all six edges can differ, computing the volume requires the Cayley–Menger determinant, a powerful technique from distance geometry. Each face is a general triangle whose area follows Heron's formula. This calculator handles both cases: enter one edge for a regular tetrahedron, or all six edges for an irregular one.
Tetrahedra appear throughout science and engineering — from molecular geometry (methane is tetrahedral) to finite-element meshes in structural analysis and 3D computer graphics. Understanding their properties is fundamental to computational geometry.
A tetrahedron calculator is most valuable when you need the full set of geometric properties, not just the volume. In design work and coursework, it is common to move between edge length, altitude, surface area, and the radii of the inscribed and circumscribed spheres. Those conversions are manageable for a regular tetrahedron, but they become much less practical once you start comparing several dimensions or checking scale changes.
It is even more useful for irregular tetrahedra because the volume is no longer a simple classroom formula. The Cayley-Menger approach depends on all six edges and is hard to evaluate accurately by hand. This tool lets you test whether an edge set forms a valid solid, compare face areas, and inspect how far an irregular tetrahedron departs from the symmetric regular case.
Regular tetrahedron (edge a): Volume: V = a³ / (6√2) Surface Area: SA = √3 × a² Face Area: √3/4 × a² Height: h = a × √(2/3) Circumradius: R = a√6 / 4 Inradius: r = a√6 / 12 Midsphere: ρ = a√2 / 4 Irregular: Volume via Cayley–Menger determinant; face areas via Heron's formula.
Result: Volume ≈ 117.85, Surface Area ≈ 173.21, Height ≈ 8.17, R ≈ 6.12, r ≈ 2.04
For a regular tetrahedron with a = 10: V = 10³ / (6√2) = 1000 / 8.485 ≈ 117.85. SA = √3 × 100 ≈ 173.21. Height = 10 × √(2/3) ≈ 8.165. R = 10√6/4 ≈ 6.124. r = 10√6/12 ≈ 2.041.
A regular tetrahedron packs several clean ratios into one solid. Once you know the edge length, every other quantity scales predictably: surface area grows with a², volume grows with a³, and the radii of the associated spheres are fixed multiples of the same edge. The identity R = 3r is especially useful because it gives a quick check that your circumradius and inradius are internally consistent.
For an irregular tetrahedron, not every collection of six edge lengths produces a real solid. Each face must satisfy the triangle inequality, and the full set must also pass the three-dimensional volume test encoded by the Cayley-Menger determinant. That is why an irregular tetrahedron can look reasonable from pairwise edge comparisons but still collapse to zero volume. The calculator helps by evaluating the geometry from the full edge set instead of relying on one face at a time.
Tetrahedra appear in structural meshes, molecular models, crystallography, and computer graphics because they are the simplest 3D building block. A mesh of tetrahedra can approximate complex volumes for simulation and analysis, and chemists use tetrahedral geometry to describe bonding arrangements such as methane. Viewing the sphere radii and face areas side by side also gives intuition for how compact or skewed a tetrahedral shape is.
A tetrahedron is a polyhedron with four triangular faces, six edges, and four vertices. It is the simplest 3D polyhedron and the 3D analogue of a triangle.
V = a³ / (6√2), where a is the edge length. For edge = 10, V ≈ 117.85 cubic units.
It is a method to compute the volume of a tetrahedron from its six edge lengths without needing coordinates. It uses a 5×5 determinant involving squared distances.
h = a × √(2/3) ≈ 0.8165 × a. For a = 10, the height is about 8.165 units.
Yes — a tetrahedron is a triangular pyramid (a pyramid with a triangular base). All four faces are triangles.
R = a√6 / 4. This is the radius of the sphere that passes through all four vertices. For a = 10, R ≈ 6.124.