perplexus.info :: Geometry : Tetrahedral Dilemma

Tetrahedral Dilemma (Posted on 2004-06-23)

Suppose you have a sphere of radius R and you have four planes that are all tangent to the sphere such that they form an arbitrary tetrahedron (it can be irregular). What is the ratio of the surface area of the tetrahedron to its volume?

Submitted by SilverKnight

Rating: 4.5000 (2 votes)

Solution: (Hide)

For each face of the tetrahedron, construct a new tetrahedron with that face as the base and the center of the sphere as the fourth vertex. Now the original tetrahedron is divided into four smaller ones, each of height R. The volume of a tetrahedron is Ah/3 where A is the area of the base and h the height; in this case h=R. Combine the four tetrahedra algebraically to find that the volume of the original tetrahedron is R/3 times its surface area.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Answer	K Sengupta	2009-01-09 11:58:33
Reference and Some Discussion	Richard	2004-06-23 23:23:19
"solution"	Charlie	2004-06-23 15:39:10
ditto	Bryan	2004-06-23 14:53:11
Another solution	Bryan	2004-06-23 14:51:45
Proof	Eric	2004-06-23 14:43:44
re: Cube	ThoughtProvoker	2004-06-23 14:36:30
Cube	Eric	2004-06-23 14:32:21

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