Volume of a regular polyhedron
Find the volume, surface area, and radii of the inscribed and circumscribed spheres of a regular polyhedron.
| Volume V | |
| Surface Area S | |
| Radius of the inscribed sphere ri | |
| Radius of the circumscribed sphere rc |
Calculation Formula: There are only five types of regular polyhedra.
| \(\normalsize Regular\ polyhedrons\\ (1)\ n:4\hspace{10px} V={\large\frac{\sqrt{2}}{12}}a^3, \hspace{40px} S=\sqrt{3}a^2, \hspace{12px} k=3\\ \hspace{25px} n:6\hspace{10px} V=a^3, \hspace{57px} S=6a^2, \hspace{20px} k=4\\ \hspace{25px} n:8\hspace{10px} V={\large\frac{\sqrt2}{3}}a^3, \hspace{42px} S=2\sqrt{3}a^2, \hspace{5px} k=3\\ \hspace{25px} n:12\hspace{5px} V={\large\frac{15+7\sqrt{5}}{4}}a^3, \hspace{5px} S=3\sqrt{25+10 \sqrt{5}}a^2, \hspace{5px} k=5\\ \hspace{25px} n:20\hspace{5px} V={\large\frac{5(3+\sqrt{5})}{12}}a^3, \hspace{5px} S=5\sqrt{3}a^2, \hspace{5px} k=3\\ (2)\ radius\ of\ insphere:\\ \hspace{40px}r_i={\large\frac{3V}{S}}\\ (3)\ radius\ of\ circumsphere:\\ \hspace{40px}r_c=\sqrt{{r_i}^2+{\large(\frac{a}{2sin{\frac{\pi}{k}}})}^2}\\ \) |
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