We are given a circle of radius $a$, and a rectangle of base $x$ is inscribed in that circle.
How to express the area of that rectangle as a function of $x$?
My Attempt:
The centroid of the rectangle (i.e. the point of intersection of the two diagonals) is the center of the circle. So each one of the two diagonals has length $2a$, which implies that the height of the rectangle is $\sqrt{ 4a^2 - x^2}$, and therefore the area is given by $$ A(x) = x \sqrt{ 4a^2 - x^2}. $$
Is this formula correct? If so, have I applied correct logic in reaching this formula? Or, have I made a mistake somewhere?