From the figure, $PQ=QR=RS=SP=42 \ m$
$\therefore$ Area of the square lawn $= \left(\text{side}\right)^2 = 42^2 = 1764 \ m^2$ $\cdots$ (1)
$\begin{aligned}
\text{Length of diagonal of the square lawn}& =SQ=PR & = & \sqrt{PQ^2 + QR^2} \\
& & & \\
& & = & \sqrt{42^2 + 42^2} = 42 \sqrt{2} \ m
\end{aligned}$
$\therefore$ Diameter of the circular flower bed $=SQ=PR=42\sqrt{2} \ m$
$\therefore$ Radius of the circular flower bed $=r=21 \sqrt{2} \ m$
$\therefore$ Area of the circular flower bed $= \pi r^2 = \dfrac{22}{7} \times \left(21\sqrt{2}\right)^2 = 2772 \ m^2$ $\cdots$ (2)
$\therefore$ Area of the circular flower bed NOT covered by the square lawn
$= 2772-1764 = 1008 \ m^2$ [from equations (1) and (2)]
$\therefore$ Area of the shaded portion $= \dfrac{1008}{2} = 504 \ m^2$