# Volume of regular polyhedrons Calculator

## Calculates the volume, surface area and radii of inscribed and circumscribed spheres of the regular polyhedrons given the side length.

number of faces n
 4 (tetrahedron) 6 (cube) 8 (octahedron) 12 (dodecahedron) 20 (icosahedron)
side length a

volume V
surface area S
radius of insphere ri
radius of circumsphere rc
 $\normal Regular\ polyhedrons\\(1)\ n:4\hspace{10} V={\large\frac{\sqrt{2}}{12}}a^3, \hspace{40} S=\sqrt{3}a^2, \hspace{12} k=3\\\vspace{5}\hspace{25} n:6\hspace{10} V=a^3, \hspace{57} S=6a^2, \hspace{20} k=4\\\hspace{25} n:8\hspace{10} V={\large\frac{\sqrt2}{3}}a^3, \hspace{42} S=2\sqrt{3}a^2, \hspace{5} k=3\\\hspace{25} n:12\hspace{5} V={\large\frac{15+7\sqrt{5}}{4}}a^3, \hspace{5} S=3\sqrt{25+10 \sqrt{5}}a^2, \hspace{5} k=5\\\hspace{25} n:20\hspace{5} V={\large\frac{5(3+\sqrt{5})}{12}}a^3, \hspace{5} S=5\sqrt{3}a^2, \hspace{5} k=3\\\vspace{5}(2)\ radius\ of\ insphere:\hspace{10}r_i={\large\frac{3V}{S}}\\(3)\ radius\ of\ circumsphere:\hspace{5}r_c=\sqrt{{r_i}^2+{\large(\frac{a}{2sin{\frac{\pi}{k}}})}^2}\\$

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