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Maximum Radius of Sphere Inside a Square Based Pyramid

For a square based pyramid with apex above the centre of the base, a sphere of maximum radius will meet the base at the centre, and each slanted face tangentially.
For the diagram below, if  
\[\]
  is the angle in the triangle ABC subtended by the radius of the circle from the midpoint of a side of the base, then  
\[tan \alpha = \frac{r}{x/2} = \frac{2r}{x}\]
.

The triangle ACD has a side AC in common, both have right angles opposite this side and radius of the sphere as another side, so are congruent. Angle DAC is the  
\[\alpha\]
  and angle DAB is  
\[2 \alpha\]
.
From the diagram,  
\[tan 2 \alpha = \frac{h}{2/x} = \frac{2h}{x}\]
.
Now use the identity  
\[tan 2 \alpha = \frac{2 tan \alpha}{1- tan^2 \alpha}\]
  to obtain
\[\frac{2h}{x} = \frac{4r/x}{1-(2r/x)^2} = \frac{4rx}{x^2 - 4r^2}\]

We can rearrange this as a quadratic in  
\[r\]
, obtaining
\[8hr^2+4rx^2-2hx^2=0\]
.
Solving this gives  
\[r=\frac{-4x^2 \pm \sqrt{16x^4 +64h^2x^2}}{16h} = \frac{-x^2 \pm x \sqrt{x^2+4h^2}}{4h}\]

Only the  
\[+\]
  option gives a valid value of  
\[r\]
  so  
\[r= \frac{-x^2 + x \sqrt{x^2+4h^2}}{4h}\]

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