The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

As shown in the diagram, rectangle $ABCD$ is a metal sheet in which $CD = 20 \, cm$ and $BC = 14 \, cm$. From it, a semicircle with diameter $\overline{BC}$ and a sector with centre $A$ and radius $AD$ is cut off. Find the area of the remaining sheet in $cm^2$.

The radius of a field in the shape of a sector is $50 \, m$. The cost of fencing its boundary is ₹ $5400$ at the rate of ₹ $30 / m$. Find the cost of tilling at the rate of ₹ $15 / m^2$. (in ₹)

In $\odot(O, 4 \, cm)$,the length of chord $\overline{AB}$ is $4 \, cm$. Then,$m \angle AOB = \ldots$ (in $^\circ$)

The radius of a circle is $3.5\,cm$. The area of the minor sector formed by two perpendicular radii of that circle is $\ldots \ldots \ldots \,cm^2$.

In $\odot(O, r)$,$\overline{OA}$ and $\overline{OB}$ are two radii perpendicular to each other. If the perimeter of the minor sector formed by these radii is $20\,cm$,then $r = \ldots\,cm$.