What expression represents the area of the square inscribed inside a circle with radius r?

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You need to find the expression for the area of the square inscribed in a circle in terms of the radius of the circle given as r.

A square inscribed in a circle is one that has the largest sides and can be fitted into the circle, because if that were not the case the corners of the square would not touch the circle.

There are two properties that you may have learnt, one of them is that a right triangle inscribed by a circle has the hypotenuse as a diameter. If you haven't learnt that, we can start with the fact that the length of the longest line segment which can be drawn in a circle is equal to its diameter. Now, the square is divided by its diagonal into two congruent right triangles each with two sides equal to the sides of the square and the hypotenuse equal to the diagonal of the square. The diagonal has the same length as the diameter of the circle.

The radius of the circle is r. The diagonal of the square is equal in length to 2r. If the side of the square is s, use the Pythagorean Theorem to get the length of the diagonal as sqrt( s^2 + s^2)

or sqrt (2*s^2) = s*sqrt 2

s*sqrt 2 = 2r

=> r = s*sqrt 2/ 2

=> r = s/sqrt 2

s = r* sqrt 2

The area of a square with side s is equal to s^2.

As s =  r*sqrt 2

=> s^2 = r^2*( sqrt 2)^2

=> s^2 = 2*r^2

Hope you understood how I got the result.

So we get the area of the square inscribed by a circle in terms of the radius of the circle as 2*r^2.

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