An example of a planar region.

A path of uniform width runs round the inside of a rectangular field $38 \mathrm{m}$ long and $32 \mathrm{m}$ width, if the path occupies $600 {\mathrm{m}}^2$ then the width of the path is

The perimeter and the breadth of a rectangle are $52 cm$ and $12 cm$ respectively. Find its area in $c{m}^2$.

Let $A_1,A_2,A_3$ be regions in the $X Y$ -plane defined by 

$A_1=\left\{(x,y):x^2+2y^2\le 1\right\}$
$A_2=\left\{(x,y):\left|x^3\right|+2\sqrt{2}|y{|}^3\le 1\right\}$
$A_3=\left\{\right(x,y):\max (\left|x\right|,\sqrt{2}\left|y\right|)\le 1\}$
Then,
 

Find the area of the shaded region in Fig. $2$, where arcs drawn with centres $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$ intersect in pairs at mid-points $\mathrm{P},\mathrm{Q},\mathrm{R} \text{and} \mathrm{S}$ of the sides $\mathrm{AB},\mathrm{BC},\mathrm{CD} \text{and} \mathrm{DA}$ respectively of a square $\mathrm{ABCD}$ of side $12 \mathrm{cm}$. $[\text{Use} \mathrm{\pi }=3.14]$

In the following figure, $\mathrm{seg} \mathrm{DH}\perp \mathrm{seg} \mathrm{EF}$ and $\mathrm{seg} \mathrm{GK}\perp \mathrm{seg} \mathrm{EF}$. If $\mathrm{DH}=6 \mathrm{cm}, \mathrm{GK}=10 \mathrm{cm}$ and $\mathrm{A}(△\mathrm{DEF})=150 {\mathrm{cm}}^2$, then find $\mathrm{A}\left(△\mathrm{DFGE}\right)$ in ${\mathrm{cm}}^2$.