Truncated icosahedron calculator
The truncated icosahedron is the famous soccer-ball solid: an Archimedean polyhedron built from 12 regular pentagons and 20 regular hexagons, with 60 vertices and 90 edges. You meet it on footballs, in the carbon-60 buckminsterfullerene molecule, and in geodesic structures. This calculator takes the single edge length, the length shared by every side of every face, and returns the two quantities that matter most: the total surface area and the enclosed volume. Because all the faces are regular polygons of the same edge length, both results scale cleanly: the surface area grows with the square of the edge and the volume with its cube, each through a fixed constant that comes from the solid's geometry. The surface area constant combines the area of 12 pentagons and 20 hexagons, while the volume constant comes from the standard formula for the solid. Enter your own edge length to size a model, a dome panel or a molecular cage, or to check a geometry problem. The results use exact square-root constants rather than rounded approximations. Every figure here is computed deterministically from the standard truncated-icosahedron formulas, shown in full below with a worked example that reconciles exactly to the calculator so you can follow each step.
For edge length a, the surface area is (20 sqrt 3 + 3 sqrt(25 + 10 sqrt 5)) a squared and the volume is (1/4)(125 + 43 sqrt 5) a cubed. At an edge of 2 units the surface area is 221.15 square units and the volume is 442.30 cubic units.
Truncated icosahedron formulas
Surface area A = ( 20 sqrt 3 + 3 sqrt(25 + 10 sqrt 5) ) a^2
Volume V = (1/4) ( 125 + 43 sqrt 5 ) a^3
a = edge length
faces = 32 (12 pentagons + 20 hexagons), edges = 90, vertices = 60
The surface area constant adds the combined area of 20 regular hexagons (the 20 sqrt 3 term) to that of 12 regular pentagons (the second term), all of edge length a. The volume constant comes from the standard closed form for the solid. Both depend only on the edge length.
Worked example
Find the surface area and volume of a truncated icosahedron with edge length a = 2.
- Surface constant = 20 sqrt 3 + 3 sqrt(25 + 10 sqrt 5) = 34.6410 + 20.6458 = 55.2868
- Surface area = 55.2868 x 2^2 = 55.2868 x 4 = 221.15 square units
- Volume constant = (1/4)(125 + 43 sqrt 5) = (1/4)(125 + 96.1437) = 55.2859
- Volume = 55.2859 x 2^3 = 55.2859 x 8 = 442.30 cubic units
- The solid has 32 faces, 90 edges and 60 vertices
The surface area is 221.15 square units and the volume is 442.30 cubic units. These are the calculator's default inputs, so the results above match the widget exactly.
Surface area and volume by edge length
| Edge a | Surface area | Volume |
|---|---|---|
| 1 | 55.29 | 55.29 |
| 2 | 221.15 | 442.30 |
| 3 | 497.58 | 1,492.77 |
Surface area scales with a squared; volume scales with a cubed.
Truncated icosahedron calculator: frequently asked questions
What is a truncated icosahedron?
A truncated icosahedron is an Archimedean solid made by slicing the 12 corners off a regular icosahedron. The result has 12 regular pentagonal faces and 20 regular hexagonal faces, 60 vertices and 90 edges. It is the shape of a classic soccer ball and of the buckminsterfullerene carbon-60 molecule.
How many faces, edges and vertices does it have?
It has 32 faces in total: 12 pentagons and 20 hexagons. It has 90 edges and 60 vertices. These satisfy Euler's formula for polyhedra, vertices minus edges plus faces equals two: 60 minus 90 plus 32 equals 2.
How does size affect surface area and volume?
Both depend only on the edge length. Surface area is proportional to the edge length squared, so doubling the edge multiplies the area by four. Volume is proportional to the edge cubed, so doubling the edge multiplies the volume by eight. The fixed constants come from the solid's geometry.
Why is it called a soccer ball?
The traditional black-and-white soccer ball is sewn as a truncated icosahedron, with black pentagons and white hexagons, then inflated so the flat faces bulge into a near sphere. The flat-faced polyhedron is the underlying geometry before inflation, which is what this calculator measures.
Are the results exact?
The formulas use exact square-root constants, so the only rounding is in the final display to two decimal places. The arithmetic is deterministic and reproducible, and the worked example reconciles exactly to the calculator output for the default edge length.
Official sources
- Geometric reference data and definitions: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.