Originally Posted by

**nickerus** 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**.

$\displaystyle a^2+(\frac{a}{2})^2=k$

where **k** is the height of one of our face triangles.

Solving, $\displaystyle k=\frac{a\sqrt{3}}{2}$

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

$\displaystyle k^2=(\frac{k}{2})^2+h^2$ ..... **<<<<< this can't be correct. See attachment.**

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