Regular Icosahedron Calculator
From one edge length, get the volume and the total surface area — the two numbers that describe any regular icosahedron.
One input, two answers
Enter the edge length and the calculator returns the volume ((5(3+√5)/12)·a³) and the surface area (5√3·a²) at once.
Keep units consistent
The edge length is unit-agnostic — your answers come back in the same unit (squared for the surface, cubed for the volume), so don't mix centimetres with inches.
What is a regular icosahedron calculator?
One edge in, full solid out
A regular icosahedron is one of the five Platonic solids: twenty identical equilateral-triangle faces meeting at twelve vertices, with thirty equal edges. Because every edge is the same length, a single measurement — the edge length a — fixes the entire shape. This calculator turns that one number into the volume (how much it holds) and the total surface area (all twenty triangles). It is the tool for d20 gaming dice, geodesic-style models, virus capsids in biology diagrams, and any geometry homework where a regular icosahedron shows up.
Enter the edge length in any length unit to get the volume and surface area instantly.
Two formulas, both built from the single edge length a — one uses the square root of 5 (≈ 2.23607), the other the square root of 3 (≈ 1.73205).
volume = (5(3 + √5) / 12) × a³The surface area is the total of twenty equilateral triangles, which adds up to 5 × √3 × a². The volume, the space inside, is (5(3 + √5) / 12) × a³: the factor 5(3 + √5)/12 is about 2.18169, so an icosahedron holds roughly two and a fifth times the cube of its edge — more than a cube of the same edge but less than a dodecahedron.
Suppose you have a regular icosahedron with an edge length of 3.
Surface area
5 × √3 × 3² = 5 × √3 × 9 = 77.942286 square units — all twenty equilateral triangles.
Volume
(5(3 + √5) / 12) × 3³ = (5(3 + √5) / 12) × 27 = 58.905765 cubic units — the space inside.
The two outputs answer two different questions. The volume (about 58.905765 cubic units for a = 3) is how much the solid holds — useful for a model, a dice mould, or a packing estimate. The single most useful insight is that the volume factor 5(3 + √5)/12 ≈ 2.18169 sits between a cube (factor 1) and a dodecahedron (factor about 7.6631): the icosahedron has the most faces of any Platonic solid yet a smaller volume-per-edge than the dodecahedron, because its triangular faces wrap more tightly. The surface area (about 77.942286 square units) is the total of all twenty triangles — what you would paint or coat, and it scales with the square of the edge, so doubling the edge quadruples the surface while the volume grows eightfold. Because both quantities grow from the same single edge, scaling the model up or down is just a matter of rescaling a, and the icosahedron's high symmetry means it has no special "top" or "front" to worry about.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Regular icosahedra and consistent units
These formulas describe a perfect regular icosahedron — twenty identical equilateral-triangle faces with every edge the same length. A non-regular or irregular twenty-faced solid (an icosahedron whose faces or edges differ in size) does not follow these formulas. The edge length is also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: an edge in centimetres gives a volume in cubic centimetres and a surface area in square centimetres, never a mix.