So we have a tetrahedral pyramid with side length a. What's the volume of the inscribed sphere that is tangent to the sides of the pyramid?
I just need some feedback on a solution, as I have no idea if it is correct. And I should mention that all the faces are equilateral triangles.
Start by solving for the height of each triangular face, with side length a.
where k is the height of one of our face triangles.
Now, we can use k to solve for the height of the pyramid, h; imagine a new triangle cutting the pyramid in half.
Using the relationship between a 2-d triangle's height and an inscribed circle's diameter, which is
we can find the diameter, d, and radius, r, of the inscribed sphere.
Plug the radius, r, into the volume of a sphere, V, to get the solution.
...and that's the solution I've come up with. And I have no idea how to even check to see if it's correct. I'll try to get a picture up so the solution is more visible.