Truncated Icosahedron
Calculator and formulas for calculating a 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.
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Truncated Icosahedron Formulas
Volume \(\small{V}\)
\(\displaystyle V=\frac{a^3 · (125+43 ·\sqrt{5}}{4}\)
Surface area \(\small{S}\)
\(\displaystyle S= 3 · a^2 · (10· \sqrt{3}+\sqrt{25+10·\sqrt{5}})\)
Outer radius \(\small{r_c}\)
\(\displaystyle r_c=\frac{a· \sqrt{58+18· \sqrt{5}}}{4}\)
Midsphere radius \(\small{r_m}\)
\(\displaystyle r_m=\frac{3 · a · (1+\sqrt{5})}{4} \)
Pentagon radius (centroid to pentagon face) \(\small{r_5}\)
\(\displaystyle r_5=\frac{a · \sqrt{\frac{1}{10}(125+41 \sqrt{5})}}{2} \)
Hexagon radius (centroid to hexagon face) \(\small{r_6}\)
\(\displaystyle r_6=\frac{a · \sqrt{\frac{3}{2} (7+3\sqrt{5})}}{2} \)
Edge length \(\small{a}\)
\(\displaystyle a= \sqrt[3]{ \frac{4 · V }{125 + 43 ·\sqrt{5}}} \)
\(\displaystyle a= \sqrt{ \frac{S}{3 ·(10· \sqrt{3}+ \sqrt{25+10 \cdot \sqrt{5})}}} \)
\(\displaystyle a=\frac{4·r_c}{\sqrt{(58+18· \sqrt{5})}}\)
\(\displaystyle a=\frac{4 · r_m}{3·(1+ \sqrt{5})} \)
\(\displaystyle a=\frac{2 · r_5}{\sqrt{\frac{1}{10}(125+41 \sqrt{5})}} \)
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