Regular tetrahedron

A regular tetrahedron is a 3-dimensional geometric solid. It is also a special type of pyramid. It consists of a base that is a triangle and a point directly over the incenter of the base, called the vertex. The edges of the tetrahedron are the sides of the triangular base together with line segments which join the vertex of the tetrahedron to each vertex of the base. All of its edges have equal measures. All faces of a regular tetrahedron are equilateral triangles.



  • Find the volume of a tetrahedron whose sides all have length $2$.



  • Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? (2007 AMC 12A Problems/Problem 20)
  • In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. Find the ratio of the volume of the smaller tetrahedron to that of the larger. (2003 AIME II, #4)