The area of a square is 144 cm$$^2$$. What is the perimeter of the square formed with the diagonal of the original square as its side?

The area of a square is 144 cm$$^2$$.

area of a square = $$side \times side$$

144 = $$\left(side\right)^2$$

side = 12

diagonal of the square = $$\sqrt{\ \left(side\right)^2\ +\ \left(side\right)^2}$$

= $$\sqrt{\ \left(12\right)^2\ +\ \left(12\right)^2}$$

= $$\sqrt{\ 144+144}$$

= $$\sqrt{288}$$

= $$12\sqrt{2}$$

perimeter of the square formed with the diagonal of the original square as its side = $$4\times 12\sqrt{2}$$

= $$48\sqrt{2}$$ cm