Largest Possible Circle in a Cube

Geometry Level 3

Given a cube of edge length s s , the radius of the largest circle that can be drawn inside the cube is given by

r = s a b r = s \sqrt{ \dfrac{a}{b} }

where a , b a,b are positive coprime integers. Find a + b a + b .


The answer is 11.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

The circle can be inscribed in a regular polygon and the largest regular polygon that can be inscribed in the cube is a regular hexagon which vertices are midpoints of the edges of the cube (one such hexagon is shown on the figure). The radius R R of the circle is the apothem O M OM of the hexagon. Hence, R = O M = 3 2 K L = 3 2 ( 1 2 E G ) = 3 2 ( 1 2 s 2 ) = s 3 8 R=OM=\frac{\sqrt{3}}{2}KL=\frac{\sqrt{3}}{2}\left( \frac{1}{2}EG \right)=\frac{\sqrt{3}}{2}\left( \frac{1}{2}s\sqrt{2} \right)=s\sqrt{\frac{3}{8}} For the answer, a + b = 3 + 8 = 11 a+b=3+8=\boxed{11} .

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