Regular tetrahedron surface and volume

Text

A regular tetrahedron has edge $12$ cm. Find the total surface area and the volume. (Take $\sqrt2\approx1.414$, $\sqrt3\approx1.732$.)

Solution

Total surface (four equilateral triangles): $S=4\cdot\frac{\sqrt3}{4}\ell^2=\sqrt3\,\ell^2=144\sqrt3\approx249.42\ \text{cm}^2.$ Volume of the regular tetrahedron: $V=\frac{\ell^3}{6\sqrt2}=\frac{1728}{6\sqrt2}=\frac{288}{\sqrt2}=144\sqrt2\approx203.65\ \text{cm}^3.$ $\boxed{S=144\sqrt3\approx249.42\ \text{cm}^2;\ V=144\sqrt2\approx203.65\ \text{cm}^3}$