Points $A$ and $B$ are distinct points on $\odot(O, r)$ and point $C$ on the circle lies in the interior of $\angle AOB$. Then,$\overline{AB} \cup \widehat{ACB}$ is ........

Circumferences of two circles are equal. Is it necessary that their areas be equal? Why?

Find the difference of the areas of a sector of angle $120^{\circ}$ and its corresponding major sector of a circle of radius $21 \, cm$. (in $cm^2$)

In a circle with radius $50 \, cm$,the area of the sector formed by a $20 \, cm$ long arc is $\ldots \ldots \ldots cm^2$.

The radius of a circular ground is $63 \, m$. Find the cost of fencing its boundary at the rate of ₹ $50 / m$. Find the cost of levelling the ground at the rate of ₹ $40 / m^2$.

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 $cm^2$).