The area of a circle is $616 \mathrm{sq}-\mathrm{m}.$ Find the area and perimeter of the square inscribed in it.

The radius of the circumcircle of a right-angled triangle is $3 \mathrm{cm}$ and the perpendicular distance of its hypotenuse from the opposite vertex is $2 \mathrm{cm}$. Find the area of the triangle.

If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

Prove that the area of the circumcircle of a square is double of the area of its incircle.