Volume of Sphere Inscribed within a Tetrahedron

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.

Solving,

Now, we can use **k** to solve for the height of the pyramid, **h**; imagine a new triangle cutting the pyramid in half.

Solving,

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.