The area of the circumscribed circle of a square of each side p units is :

1. 2πp² sq units

2. $\dfrac{πp^2}{4}$ sq units

3. $\dfrac{πp^2}{2}$ sq units

4. πp² sq units

ABCD is a square with diagonal 'd' units and side 'p' units.

From figure,

Diameter of the circle = diagonal of the square.

Side of square = p units.

$\text{Diagonal of square (d)} = \sqrt{p^2 + p^2} \\[1em] = \sqrt{2p^2} \\[1em] = p\sqrt{2}$

Radius of circle = $\dfrac{d}{2} = \dfrac{p\sqrt{2}}{2}$ units.

Calculating,

$\text{Area of circle} = πr^2 \\[1em] = π\Big(\dfrac{p\sqrt{2}}{2}\Big)^2 \\[1em] = π \times \dfrac{p^2 × 2}{4} \\[1em] = \dfrac{πp^2}{2} \text{ sq units}.$

Hence, option 3 is the correct option.