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

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.