## 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 \[r= \frac{-x^2 + x \sqrt{x^2+4h^2}}{4h}\]