Radius of each circular design $=r=7 \ cm$
$\therefore$ Diameter of each circular design $= 14 \ cm$
$\therefore$ Length of side of the square handkerchief $=s= 14 \times 3 = 42 \ cm$
$\therefore$ Area of the square handkerchief $=s^2 = 42^2 = 1764 \ cm^2$ $\cdots$ (1)
Area of each circular design $=\pi r^2 = \dfrac{22}{7} \times 7^2 = 154 \ cm^2$
$\therefore$ Area of circular designs $= 9 \times 154 = 1386 \ cm^2$ $\cdots$ (2)
$\therefore$ Required area [from equations (1) and (2)] $=1764-1386=378 \ cm^2$