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

In a circle with radius $10\,cm$,the area of a minor sector is $40\,cm^2$. Then,the length of the arc corresponding to that sector is $\ldots \ldots \ldots \ldots \,cm$.

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 ₹)

The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii $24 \, cm$ and $7 \, cm$ is (in $cm$):

In $\odot(O, r)$,the length of minor $\widehat{ACB}$ is one-eighth of the circumference of the circle. Then,the measure of the angle subtended at the centre by that arc is $\ldots \ldots \ldots \ldots$ (in $^\circ$)

In the figure,$ABCD$ is a trapezium with $AB \parallel DC$,$AB = 18 \, cm$,$DC = 32 \, cm$,and the distance between $AB$ and $DC = 14 \, cm$. If arcs of equal radii $7 \, cm$ with centers $A, B, C$,and $D$ have been drawn,find the area of the shaded region of the figure (in $cm^2$).