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Solution

The square is inscribed in the circle. So the diagonal of the square = diameter of circle = $2r$. Let side of square be $a$. Using diagonal formula: $ a\sqrt{2} = 2r $ $ a = \dfrac{2r}{\sqrt{2}} = r\sqrt{2} $ Area of square: $ a^2 = (r\sqrt{2})^2 = 2r^2 $ The shaded region is exactly half of the square (a triangular half). So shaded area = $ \dfrac{1}{2} \times 2r^2 = r^2 $