Side length of a regular tetrahedron given the radius of the sphere tangent to its edges

Question

I am working on a physics project where we have to build a container to protect a glass ornament when dropped from a high place. My design involves building a tetrahedron out of straws and putting the glass ornament inside of it. I'm not sure what length to cut each side of the straw tetrahedron. I did some measurements and I estimated the radius of the sphere to be about 1.3. The tetrahedron has no "walls" where the ornament will touch at a single point. It is simply an empty frame made out of regular McDonald's straws, and the ornament will protrude out of each side a little bit, thus there are no faces, only edges. How can I find the best side length?

Answer 1

Added: in the construction below, the point on an edge closest to the origin is at $(0,0,1),$ radius for this is exactly $1,$ therefore the edge length divided by $\sqrt 8$

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one way to place a regular tetrahedron is at every other vertex of the cube with all vertices $(\pm1,\pm1,\pm1).$ For example $$ (-1,-1,-1) \; , \; \; \; (-1,1,1) \; , \; \; \; (1,-1,1) \; , \; \; \; (1,1,-1) \; . \; \; \; $$ The edges are all length $\sqrt 8$

One triangle side is $$ x+y+z=1. $$ The closest point to the origin in that plane is $$ \left( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right) $$ with distance from the origin $$ \frac{\sqrt 3}{3} = \frac{1}{\sqrt 3},$$ which is the radius of the inscribed sphere. Thus the radius is the edge length divided by $$ \sqrt{24} $$

Answer 2

Your inscribed sphere is touching edges of tetrahedron, not its faces which do not exist in reality. The radius of the sphere touching edges of terahedron is known to be:

$$r=\frac{a}{\sqrt8}\iff a=2r\sqrt2\approx2.82r$$

Proving the formula is a fairly simple exercise.

https://en.wikipedia.org/wiki/Tetrahedron#Formulas_for_a_regular_tetrahedron