ratio of area of circles

Geometry Level 2

In the diagram, a large circle is inscribed in a square. A small circle is then inscribed in a corner, touching the large circle and the sides of the square. How many times larger is the radius of the large circle than that of the small circle?


The answer is 5.828.

Harsh Patel by Harsh Patel , 25, Kenya

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1 solution

Let the radius of the large circle be 1. By pythagorean theorem, B D = 2 BD=\sqrt{2} .

Let the radius of the small circle be H E = G E = F E = r HE=GE=FE=r . By pythagorean theorem, D E = r 2 DE=r\sqrt{2} .

Then,

B D = B F + F E + D E BD = BF+FE+DE

2 = 1 + r + r 2 \sqrt{2}=1+r+r\sqrt{2}

r = 2 − 1 1 + 2 ≈ 0.171573 r=\dfrac{\sqrt{2}-1}{1+\sqrt{2}} \approx 0.171573

The desired answer is 1 0.171573 ≈ 5.828 \dfrac{1}{0.171573} \approx \boxed{5.828}

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