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Tetrahedron bounded by spheres (Posted on 2003-04-17) Difficulty: 3 of 5
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?

See The Solution Submitted by Bryan    
Rating: 3.5000 (6 votes)

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Solution (long and complicated version) | Comment 1 of 7
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
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