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?

so you want the distance from the center point, (C), to the center of one of the sides, this will be the shorter, (r). the distance from C to a vertex where three planes meet will be (R).

take any two planes of the tetrahedron and draw a line perpendicular to the center point of a triangle (t). Now you will have two lines that intersect. the intersection point is (C) of the tetrahedron.

line Ct = r; line C to the vertex = R

knowing that all of the angles in the tetrahedron are 60*, half would be 30*. the intersection at t is known to be 90*. making a 30-60-90 triangle.

in such a triangle we know that the hypotenus(?) to the short leg is 2:1. in this case R = the hypoteneus(?) and r = the short leg.

So 2:1

Thank goodness I'm not a teacher

Posted by Hank on 2003-04-17 09:30:41