Tetrahedron Calculator
Calculate the volume, surface area, and height of a regular tetrahedron from its edge length.
🔻 What is a Regular Tetrahedron?
A regular tetrahedron is a three-dimensional solid made of 4 identical equilateral triangle faces, 6 equal edges, and 4 vertices where exactly 3 faces meet. It is the simplest of the five Platonic solids, the only one with just 4 faces, and every one of its edges shares the exact same length, usually written as a. Because all edges are equal, a single measurement, the edge length, is enough to fully determine the tetrahedron's volume, surface area, height, and every other dimension.
Tetrahedral shapes show up throughout science and engineering. In chemistry, the methane molecule (CH4) has tetrahedral molecular geometry, with the carbon atom at the centre and its 4 hydrogen atoms positioned at the vertices of a regular tetrahedron, a pattern repeated in countless other molecules with four-fold bonding. In games, the 4-sided die (d4) used in tabletop role-playing games is a physical tetrahedron. In structural engineering and architecture, tetrahedral and other triangulated space-frame trusses are prized for rigidity, since a tetrahedron is the simplest 3D shape that cannot flex or distort without changing an edge length. Crystallography also uses tetrahedral coordination to describe how atoms bond in many mineral and semiconductor structures.
A common point of confusion is the difference between "tetrahedron" and "pyramid." Every tetrahedron is technically a triangular pyramid, since it has a triangular base and three triangular sides meeting at an apex. However, in everyday language "pyramid" usually means a square-based pyramid like the Great Pyramid of Giza, which has one square base and four triangular sides, a different shape from a regular tetrahedron where all four faces, including the base, are identical equilateral triangles.
This calculator computes a regular tetrahedron's volume, surface area, height (from any vertex straight down to the centre of the opposite face), circumradius (centre to a vertex), and inradius (centre to a face) from just the edge length, with a labelled wireframe diagram showing which edge is being measured. It is useful for geometry students, chemistry students visualising molecular shapes, and anyone estimating material or volume for a tetrahedral structure or container.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Small Tetrahedron
Edge length a = 5 units
Example 2 — Larger Tetrahedron
Edge length a = 10 units
Example 3 — Small Model or Die (d4)
Edge length a = 3.5 units (a typical small tetrahedral game die)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for the volume of a regular tetrahedron?
Volume = a³ / (6√2), where a is the edge length. For an edge length of 5 units, Volume = 125 / 8.4853 ≈ 14.73 cubic units. All six edges of a regular tetrahedron are equal, so only one measurement is needed.
What is the formula for the surface area of a regular tetrahedron?
Surface Area = √3 × a², where a is the edge length, since all 4 faces are equilateral triangles each with area (√3/4)a². For a = 5, Surface Area = √3 × 25 ≈ 43.30 square units.
What is the height of a regular tetrahedron?
Height (apex to the centre of the opposite base) = a × √(2/3) = a × √6 / 3, where a is the edge length. For a = 5, Height = 5 × 0.8165 ≈ 4.08 units. This is the perpendicular distance from any vertex straight down to the centroid of the triangular face opposite it.
What is a regular tetrahedron?
A regular tetrahedron is a solid with 4 faces, all of which are congruent equilateral triangles, 6 equal edges, and 4 vertices where 3 faces meet. It is the simplest of the five Platonic solids and the only Platonic solid with just 4 faces.
What is the difference between a tetrahedron and a pyramid?
Every tetrahedron is technically a triangular pyramid (4 triangular faces), but 'pyramid' in everyday use usually refers to a square-based pyramid like the ones in Egypt, which has one square base and 4 triangular sides. A regular tetrahedron specifically has all 4 faces as identical equilateral triangles, not a square base.
What is the circumradius of a regular tetrahedron?
Circumradius (centre to any vertex) = a × √6 / 4, where a is the edge length. For a = 5, circumradius ≈ 3.06 units. This is the radius of the sphere that passes through all 4 vertices.
What is the inradius of a regular tetrahedron?
Inradius (centre to the middle of any face) = a × √6 / 12, where a is the edge length. For a = 5, inradius ≈ 1.02 units. This is the radius of the largest sphere that fits entirely inside the tetrahedron, touching all 4 faces. Notice the circumradius is always exactly 3 times the inradius.
Where do tetrahedron shapes appear in real life?
Tetrahedral shapes appear in chemistry (the tetrahedral molecular geometry of methane, CH4, with the carbon at the centre and 4 hydrogens at the vertices), in some tetrahedral dice used in tabletop games (4-sided d4 dice), in certain space-frame and truss structures in architecture for rigidity, and in crystallography for describing atomic bonding arrangements.
Why is the tetrahedron the most rigid basic 3D shape?
A tetrahedron is the simplest 3D shape that cannot be deformed without changing at least one edge length, because every vertex is directly braced by triangles to every other vertex. This is why triangulated tetrahedral trusses are common in bridges, towers, and space frames where structural rigidity with minimal material is important.
How many faces, edges, and vertices does a tetrahedron have?
A tetrahedron has 4 faces, 6 edges, and 4 vertices. This satisfies Euler's formula for polyhedra: Vertices − Edges + Faces = 4 − 6 + 4 = 2, the same relationship that holds for a cube (8 − 12 + 6 = 2) and every other convex polyhedron.
Does the volume formula work for an irregular tetrahedron?
No. This calculator's formulas (a³/6√2 for volume, √3a² for surface area) assume a regular tetrahedron where all 6 edges are equal length and all 4 faces are congruent equilateral triangles. An irregular tetrahedron, with four different-length edges, needs a more general formula based on vertex coordinates, such as the Cayley-Menger determinant.