I have a question that I've been working upon for a long time but in vain. Can you help me.
Determine the relation between the volume of a con circumscribed to a regular tethadron and the volume of a sphere inscribed in tethadron.
What I tried to do is:!
You'll help me a lot! Thank you !
On
It's very hard to understand what you're writing, because there're no words at all. Can you do the following steps:
1) Find the height of the tetraedron as a function of its side
2) Find the radius if inscribed sphere as a function of the side of the tetraedron
3) Find the radius of the circle forming the base of the cone - still as a function of the side of the tetraedron
4) Find the volume of the cone
5) Find the volume of the inscribed sphere
6) Conclude.
If you have questions regarding any of the above steps, ask in comments.
Working quickly, with tetrahedral edges 1 unit long, I found the base of the cone as $1\over\sqrt{3}$, the height of the cone as $\sqrt{2\over 3}$, and the radius of the sphere as $1\over 2\sqrt{3}$. If your answers differ, I'll redo it more carefully.