A wire, when bent in the form of a square, encloses an area of 484 cm². Find :

(i) one side of the square

(ii) length of the wire

(iii) the largest area enclosed, if the same wire is bent to form a circle.

Given:

Area of the square = 484 cm²

Let s be the side of the square.

As we know, the area of the square = side²

⇒ s² = 484

⇒ s = $\sqrt{484}$

⇒ s = 22

Hence, one side of square is 22 cm.

(ii) Total length of the wire = perimeter of the square

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

= 4 x 22

= 88 cm

Hence, the length of the wire is 88 cm.

(iii) Perimeter of the square = circumference of the circle

Let r be the radius of the circle.

⇒ 2πr = 88 cm

$⇒ 2 \times \dfrac{22}{7} \times r = 88\\[1em] ⇒ \dfrac{44}{7} \times r = 88\\[1em] ⇒ r = \dfrac{7 \times 88}{44}\\[1em] ⇒ r = \dfrac{616}{44}\\[1em] ⇒ r = 14 \text{ cm}$

Area of the circle = πr²

$= \dfrac{22}{7} \times 14^2\\[1em] = \dfrac{22}{7} \times 196\\[1em] = \dfrac{4312}{7}\\[1em] = 616 \text{ cm}^2$

Hence, the largest area that can be enclosed when the same wire is bent to form a circle is 616 cm².