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Puzzle #16

Two concentric circles, a small one and a big one. A square is inscribed into the big circle and the small circle is inscribed into the square.

An annulus has area 4π.

What is the area of the square?

Solution

Let r and R be the radii of the small and big circles, respectively.

Two concentric circles, a small one and a big one. A square is inscribed into the big circle and the small circle is inscribed into the square. The radius of the small circle, from the center/centre to the right side of the square, is labeled/labelled r and the radius of the big circle, from the center/centre and the right top vertex of the square, is labeled/labelled R.

As the area of the annulus is 4π, we get

\pi\left(R^2-r^2\right)=4\pi

R^2-r^2=4

r^2=R^2-4\; \boldsymbol{\left(1\right)}

Now, consider the isosceles right-angled triangle inside the square. It has sides with lengths r, r and R.

Using the Pythagoras’ theorem, we have

R^2=r^2+r^2

R^2=2r^2\; \boldsymbol{\left(2\right)}

Substituting the value of (2) in (1) we get

r^2=2r^2-4

r^2=4

r=2

Then, as the side of the square has length 2r,

2r=4

\left(2r\right)^2=16

So, the area of the square is 16.

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