Inscribed and circumscribed spheres of a regular tetrahedron

Text

A regular tetrahedron has edge $6$ cm. Find the radius of the circumscribed sphere and the radius of the inscribed sphere. (Take $\sqrt6\approx2.449$.)

Solution

For a regular tetrahedron of edge $\ell$: $R=\frac{\ell\sqrt6}{4}=\frac{6\sqrt6}{4}=\frac{3\sqrt6}{2}\approx3.67\ \text{cm},\qquad r=\frac{\ell\sqrt6}{12}=\frac{6\sqrt6}{12}=\frac{\sqrt6}{2}\approx1.22\ \text{cm}.$ (Note that $R=3r$.) $\boxed{R=\tfrac{3\sqrt6}{2}\approx3.67\ \text{cm};\ r=\tfrac{\sqrt6}{2}\approx1.22\ \text{cm}}$