If the sum of the circumferences of two circles with radii $R_{1}$ and $R_{2}$ is equal to the circumference of a circle of radius $R$,then

$A$ horse is tied to a peg at one corner of a square-shaped grass field of side $15 \, m$ by means of a $5 \, m$ long rope (see figure). Find:
$(i)$ The area of that part of the field in which the horse can graze.
$(ii)$ The increase in the grazing area if the rope were $10 \, m$ long instead of $5 \, m$. (Use $\pi = 3.14$)

In a circle with radius $21 \, cm$,the length of a minor arc is $33 \, cm$. Find the measure of the angle subtended at the centre by this arc. Also,find the area of the minor sector and the area of the minor segment formed by it.

Fig. depicts an archery target marked with its five scoring regions from the centre outwards as $Gold, Red, Blue, Black$ and $White.$ The $diameter$ of the region representing Gold score is $21\, cm$ and each of the other bands is $10.5 \,cm$ wide. Find the area of each of the five scoring regions. [use $\pi=\frac{22}{7}$]

Find the area of the sector of a circle with radius $4\, cm$ and of angle $30^{\circ}$. Also,find the area of the corresponding major sector (in $cm^2$) (Use $\pi = 3.14$).

The length of the minute hand of a clock is $10.5 \, cm$. The area of the region swept by it in $20$ minutes is $\ldots \, cm^2$.