Math CentralQuandaries & Queries


Question from George, a parent:

A circle within a square which is inside a larger circle which is also within a square. (a circle in a square inside a circle in a square)
Equation of the smaller circle is: x ^ 2 x y ^ 2 = 25.
What are the dimensions of the larger square?
Been 40 years, trying to help my son.

Hi George,

I think the equation of the smaller circle is $x^2 + y^2 = 25.$

A circle with centre at the origin and radius $r$ has equation $x^2 + y^2 = r^2$ and thus the smaller circle has radius 5 units. Here is my diagram. $C$ is the origin.

squares and circles

Triangle $ABC$ is a right triangle and $|AB| = |BC| = 5.$ Use Pythagoras' theorem to find the length of $CA.$ This is the radius of the larger circle.

Can you complete the problem now?

Rather than asking for the dimensions of the larger square I would have asked for the length of its diagonal. Can you find the length of the diagonal?


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