Area of a sector of central angle $200^{\circ}$ of a circle is $770 \, cm^{2}$. Find the length of the corresponding arc of this sector. (in $cm$)

$A$ calf is tied with a rope of length $6 \,m$ at the corner of a square grassy lawn of side $20 \,m$. If the length of the rope is increased by $5.5 \,m$,find the increase in area of the grassy lawn in which the calf can graze. (in $m^2$)

In a circular table cover of radius $32 \, cm$,a design is formed by leaving an equilateral triangle $ABC$ in the middle as shown in the figure. Find the area of the design. [Use $\pi = \frac{22}{7}$]

An archery target has three regions formed by three concentric circles as shown in the figure. If the diameters of the concentric circles are in the ratio $1: 2: 3$,then find the ratio of the areas of the three regions.

In the adjoining figure,$PS$ is the diameter of a circle and $PS = 12$. $PQ = QR = RS$. Semicircles are drawn with diameters $\overline{PQ}$ and $\overline{QS}$. Find the perimeter and the area of the shaded region. $(\pi = 3.14)$

$A$ piece of wire $20 \,cm$ long is bent into the form of an arc of a circle subtending an angle of $60^{\circ}$ at its centre. Find the radius of the circle. (in $cm$)