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Sphere in Tetrahedron (Posted on 2005-04-19) Difficulty: 4 of 5
Create a tetrahedron starting with a square ABCD (clockwise), add 2 points E and F placing them halfway between AB and AD respectively.

Fold along lines EF, EC and FC.

Then, form the tetrahedron by keeping triangle EFC as the base, and folding triangles AEF, BEC, and DFC up so that points A, B, and D all meet.

For AB=a find the radius of the largest sphere contained by this tetrahedron

No Solution Yet Submitted by joe    
Rating: 2.5000 (2 votes)

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Solution Puzzle Solution Comment 7 of 7 |
(In reply to Answer by K Sengupta)

we know that the volume of a tetrahedron (V) is given by:

V = Ah/3, and:
V = S*r/3
where: A = base area
h = height
S = surface area
r = inradius

Then, Sr = Ah
-> r = Ah/s

Now, FD = a, considering triangle FDC as base, we have:

(A, h, s) = (a^2/8, a, a^2), so that:
r = (a^3)/(8*a^2) = a/8

Consequently, the required radius of the largest sphere contained in the tetrahedron is a/8.

Edited on December 16, 2008, 1:34 pm
  Posted by K Sengupta on 2008-12-16 04:19:31

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