How to find the area of a circle with a square inside with its side length given?

A square that fits snugly inside a circle is inscribed in the circle. The square's corners will touch, but not intersect, the circle's boundary, and the square's diagonal will equal the circle's diameter. Also, the square's diagonal is equal to the hypotenuse of a right-angled triangle, which is also equal to the diameter of the circle.

Answer: Using the side length of the square, find the diagonal (and hence the diameter of the circle) of the square, and then apply the area formula of the circle to find its area.

Let's understand the solution.

Explanation:

Let's consider the above figure in which a circle encloses a square according to the given question.

Let's assume the length of the side of the square to be 10 units.

⇒Then, by using the Pythagoras theorem, we get the length of the diagonal, which is also the diameter of the circle (according to the properties of circle).

⇒Therefore, diameter of circle = diagonal of square = √(10² + 10²) = 10√2 units

⇒Hence, the radius = diameter / 2 = 5√2 units

⇒Now, the area of the circle = π (radius)² = 50π sq. units

Hence, by this example, we understand that using the side length of the square, we find the diameter (and hence the radius) of the circle, and then apply the area formula of the circle to find its required area.