A wire, when bent in the form of a square, encloses an area of 196 cm². If the same wire is bent to form a circle, find the area of the circle.

Given:

Area of the square = 196 cm²

Let s be the side of the square.

As we know, the area of the square = side²

⇒ s² = 196

⇒ s = $\sqrt{196}$

⇒ s = 14

As we know, the perimeter of the square = 4 x side

= 4 x 14

= 56 cm

Perimeter of the square = circumference of the circle

Let r be the radius of the circle.

⇒ 2πr = 56

$⇒ 2 \times \dfrac{22}{7} \times r = 56\\[1em] ⇒ \dfrac{44}{7} \times r = 56\\[1em] ⇒ r = \dfrac{7 \times 56}{44}\\[1em] ⇒ r = \dfrac{392}{44}\\[1em] ⇒ r = 8\dfrac{10}{11} \text{ cm}$

Area of the circle = πr²

$= \dfrac{22}{7} \times \Big(8\dfrac{10}{11}\Big)^2\\[1em] = \dfrac{22}{7} \times \Big(\dfrac{98}{11}\Big)^2\\[1em] = \dfrac{22}{7} \times \Big(\dfrac{9604}{121}\Big)\\[1em] = \dfrac{211,288}{847}\\[1em] = 249.45 \text{ cm}^2$

Hence, the area of the circle is 249.45 cm².