# 199 geometry/tetrahedron.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 199 geometry/tetrahedron.p

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?

geometry/tetrahedron.s

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.

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