Truncated Icosahedron

This function calculates various properties of a truncated icosahedron. A truncated icosahedron is created by cutting off the corners of a dodecahedron so that all edges are the same length. It is a polyhedron with 32 sides, 90 edges and 60 vertices. They form 12 regular pentagons and 20 regular hexagons.

To perform the calculation, select the property you know and enter its value. Then click on the 'Calculate' button.

Truncated Icosahedron Calculator  Input Delete Entry Edge length Volume Surface area Outer radius Midsphere radius Decimal places 0 1 2 3 4 6  Results Edge lengtha VolumeV Surface areaS Outer radius rc Midsphere radius rm Pentagon radius r5 Hexagon radius r6

Formulas for truncated icosahedrons

Volume

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

\(\displaystyle a= \sqrt[3]{ \frac{4 · V }{125 + 43 ·\sqrt{5}}} \)

Surface area

\(\displaystyle S= 3 · a^2 · (10· \sqrt{3}+\sqrt{25+10·\sqrt{5}})\)

\(\displaystyle a= \sqrt{ \frac{S}{3 ·(10· \sqrt{3}+\sqrt{25+10·\sqrt{5})}}} \)

Outer radius

\(\displaystyle rc=\frac{a· \sqrt{58+18· \sqrt{5}}}{4}\)

\(\displaystyle a=\frac{4·rc}{\sqrt{(58+18· \sqrt{5})}}\)

Midsphere radius

\(\displaystyle rm=\frac{3 · a · (1+\sqrt{5})}{4} \)

\(\displaystyle a=\frac{4 · rm}{3·(1+ \sqrt{5})} \)

Pentagon radius (centroid to pentagon face)

\(\displaystyle r5=\frac{a · \sqrt{\frac{1}{10}(125+41 \sqrt{5})}}{2} \)

\(\displaystyle a=\frac{2 · r5}{\sqrt{\frac{1}{10}(125+41 \sqrt{5})}} \)

Hexagon radius (centroid to hexagon face)

\(\displaystyle r6=\frac{a · \sqrt{\frac{3}{2} (7+3\sqrt{5})}}{2} \)

\(\displaystyle a=\frac{2 · r6}{\sqrt{\frac{3}{2} (7+3\sqrt{5})}} \)