Circle inscribed in a square

Question

In a square with side a is inscribed a cirle. In the same circle is inscribed another square. Find the sum of the radiuses of the inscribed circles.

*I suppose that we would use the geometrical progression and the side of the big square is equal to the diameter of the first circle. Then the side of the smaller square is equal to the radius of the biq square and etc.

Please, give me some directions how to solve this problem

Answer 1

The diagonals of a square intersect at right angles. The radius of the circle is a/2. If ABCD is the inscribed square and the centre of the circle is O, then AO=BO=CO=DO=a/2 and ∠AOB=90°.

Therefore AB=BC=CD=AD=√(a²/4+a²/4)=a/√2 or a√2/2.

The circle inscribed in square ABCD has diameter=AB, so its radius is AB/2=a√2/4.

The sum of the radii is a/2+a√2/4=0.8536 approx.

This can be written as a(2+√2)/4.