A regular tetrahedron holds a sphere snugly within its four sides. A larger sphere surrounds the tetrahedron, just touching its four vertices. What is the ratio of radii of the two spheres?

Erect the medians (which are also angle bisectors) on each of the four faces. Choose one face as the base. A vertical line through the meeting point of the medians (the center, equidistant from each vertex and each side) of the base will pass through the centers of the inscribed and circumscribed spheres, as will any line drawn perpendicular to any of the faces at their centers.

The center of the base divides each median line in a 1 to 2 ratio—one unit away from the edge and two units from the vertex. (Let’s keep that unit of measure: the distance from the center of the triangle to the edge.) The one-unit segment from the center of the base to one of the edges, together with the median of the adjacent face at that edge, and the vertical line from the center of the base to the top vertex of the tetrahedron, forms a right triangle, with the 1-unit segment on the base being the shorter leg and the median going up the adjacent face being the hypotenuse. That median has length 1+2=3, just like all the other medians. Call this the median of the second face.

A smaller right triangle can be constructed within this larger one. This time the longer leg will be the portion of the median extending from the top vertex of the tetrahedron, down the slant-height median to the center of the second face. The short leg is the perpendicular to the second face at its center extending to the center of the tetrahedron. This right triangle is similar to the larger one as they share their acute angle, but the hypotenuse is now along the vertical centerline, extending from the top vertex to the center of the tetrahedron (as the perpendiculars from each of the centers of the faces meet there as stated previously). The short leg is the radius of the inscribed sphere and the hypotenuse is the radius of the circumscribed sphere. As this smaller triangle is similar to the larger one, the ratio of the short leg to the hypotenuse is again 3, which is now seen to be the ratio of the radii of the two spheres: 3:1.

Posted by Charlie on 2003-04-17 09:42:15