Amazing circle and square puzzle.

Question: If the red area (between a square and its inscribed circle) equals the blue area (between a smaller circle and its inscribed square), what is the value of:

(large circle’s radius)/(small circle’s radius)

Answer To Inscribed Circles And Squares Puzzle

Suppose the large circle has radius R and the small circle has radius r.

Red area
The large square has a side length equal to 2R. Therefore,

Area 1 = area large square – area large circle

Area 1 = (2R)2 – π(R)2

Area 1 = 4R2 – πR2

Area 1 = R2(4 – π)

Blue area
The diameter of the small circle is the diagonal of the small square, so the square has a diagonal equal to 2r, meaning its side is equal to 2r/√2 = r√2.

Area 2 = area small circle – area small square

Area 2 = πr2 – (r√2)2

Area 2 = πr2 – 2r2

Area 2 = r2(π – 2)

So that the areas are equal, we need:

Area 1 = Area 2

R2(4 – π) = r2(π – 2)

R2/r2 = (π – 2)/(4 – π)

R/r = √[(π – 2)/(4 – π)] ≈ 1.153

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