Truncated Icosahedron Calculator

Name: Truncated Icosahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular truncated icosahedron. A truncated icosahedron is constructed by cutting off the vertices of an regular icosahedron in a way, so that every edge has the same length. Hereby, the vertices of the icosahedron become 12 regular pentagons and the faces become 20 regular hexagons. The dual solid of the truncated icosahedron, this is the shape that results when the midpoints of the connected faces of the truncated icosahedron are connected by edges, is the pentakis dodecahedron.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

a^{'} = 3 a

A = 3 a^{2} \cdot (10 \sqrt{3} + \sqrt{25 + 10 \cdot \sqrt{5}})

V = \frac{a^{3}}{4} \cdot (125 + 43 \cdot \sqrt{5})

r_{c} = \frac{a}{4} \cdot \sqrt{58 + 18 \cdot \sqrt{5}}

r_{m} = \frac{3}{4} a \cdot (1 + \sqrt{5})

Volume diagonal = 2 \cdot r_{c}

The truncated icosahedron is an Archimedean solid. Edge length and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The first known image of a truncated icosahedron appeared in 1460 in a book by the Italian painter Piero della Francesca, many centuries after Archimedes.
This shape is known, among other things, as a soccer ball shape. It got this name because soccer balls, although spherical in shape, are often composed of several regular pieces. The 1970 Soccer World Cup in Mexico saw the first use of the soccer ball shape that is still widely used today. It consists of rounded pentagonal and hexagonal pieces that, if flat, would form a truncated icosahedron.
Another structure in the shape of a truncated icosahedron is the C₆₀ molecule Buckminster fullerene, also called a buckyball, which is made of 60 carbon atoms. This is an artificial modification of the element carbon and was named after the architect Richard Buckminster Fuller, who built geodesic domes with similar shapes. Buckminster fullerenes are suitable, for example, for use in organic solar cells.

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