The regular tetrahedron

The regular tetrahedron is the most symmetric pyramid: a pyramid with a triangular base and all faces equal.

Example — Regular tetrahedron

All the faces are equilateral triangles with side $\ell=2$.

• Height of the equilateral triangle of the base: $h_b=\sqrt{3}$.
• Apothem of the base: $m = h_b/3 = \sqrt{3}/3$.
• Radius of the circle inscribed in the base: $r = \sqrt{3}/3$.
• Height of the tetrahedron: $h=\sqrt{\ell^2 - (2m)^2} = \sqrt{4 - \tfrac{4}{3}} = \sqrt{\tfrac{8}{3}} = 2\sqrt{\tfrac{2}{3}}.$

Links

Topics: Synthetic geometry of space
Concepts: Apothem · Pyramid · Regular polyhedra
Skills: Calculating · Using formulas