3450.21 – Tetrahedron Inscribed in a Cube
Four of the eight vertexes of a cube are the vertexes of a regular tetrahedron as in the figure below. Relax for a moment and enjoy how neatly the regular tetrahedron fits inside the cube. The other four vertexes of the cube, incidentally, form a second, inter-penetrating tetrahedron.
Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
Let the edge of the cube be . Then , the surface area of the cube, is .
The faces of the tetrahedron are equilateral triangles whose edges are diagonals of the cube's square faces. If is the edge length of the tetrahedron, then since the diagonal of a square's face is the hypotenuse of a 45-45-90 triangle. Now the area of an equilateral triangle of edge is so , the surface area of the tetrahedron, is
Thus the desired ratio of areas is
Students who like making models might make a nice 3-d model of this figure.
(N.b. the 45-45-90 right triangle is sometimes known as the western, or cowboy's, triangle. Why?)