Prove that the rectangle of greatest area could be inscribed in a circle is a square and find the area

in terms of the radius.

Start the assumption as it is a square of length X and width Y and a radius R

The circle is given as x² + y² = r², y = sqrt(r² - x²)
The area of the rectangle is going to be
A = 2x * 2y = 4x * sqrt(r²-x²)

dA/dx = 4sqrt(r²-x²) - [ 4x² / sqrt(r²-x²) ]
= 4r²-8x² / sqrt(r²-x²) = 0

Hence, 2x² = r², x = r / √2, 2x = 2y = √2(r)