Circle in a square in a circle problem

Geometry Level 2

Edge of the unit square on the picture is in the middle of a bigger circle. Two of its sides are tangent to both circles. Find the radius of the smaller circle.

1 − 2 2 1 - \frac{{\sqrt 2 }}{2} 2 − 1 \sqrt 2 - 1 ( 1 − 2 ) 2 {\left( {1 - \sqrt 2 } \right)^2} 2 4 \frac{{\sqrt 2 }}{4} 1 4 \frac{1}{4}

Tomáš Hauser by Tomáš Hauser , 21, Czech Republic

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1 solution

Marta Reece
Aug 1, 2018

The diagonal of the square is 1 + R ( 1 + 2 ) = 2 1+R(1+\sqrt2)=\sqrt2

The solution of this is R = 3 − 2 2 = ( 1 − 2 ) 2 R=3-2\sqrt2=\boxed{(1-\sqrt2)^2}

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