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- Feb 14, 2012
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Consider a pyramid whose base is an $n$-gon with side length $s$, and whose height is $h$. What is the radius of the largest sphere that will fit entirely within the pyramid?
Hi MarkFL, thanks for participating and your solution is so smart and short!Here is my solution:
Consider a cross-section of the pyramid-sphere system from the axis of symmetry running along an apothem $a$ of the base:
View attachment 1083
Now, orienting the right angle at the origin of the $xy$-plane, we find that the equation of the line along which the hypotenuse runs is:
\(\displaystyle y=-\frac{h}{a}x+h\)
We require that the perpendicular distance from the center of the circle to this line be $r$, hence:
\(\displaystyle r=\frac{|h-r|}{\sqrt{\left(\frac{h}{a} \right)^2+1}}\)
Since $h>r$, and solving for $r$, we find:
\(\displaystyle r=\frac{a\left(\sqrt{h^2+a^2}-a \right)}{h}\)
Now, the apothem $a$ is given by:
\(\displaystyle a=\frac{s}{2}\tan\left(\frac{\pi(n-2)}{2n} \right)\)