Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii $15 \, cm$ and $18 \, cm$ (in $cm$).

The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters $36 \, cm$ and $20 \, cm$ is (in $cm$)

The areas of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement true? Why?

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 $\ldots \, cm^2$.

In the figure,arcs are drawn by taking vertices $A, B$ and $C$ of an equilateral triangle of side $10 \, cm$ as centers to intersect the sides $BC, CA$ and $AB$ at their respective mid-points $D, E$ and $F$. Find the area of the shaded region (Use $\pi = 3.14$) (in $cm^2$).

As shown in the diagram,$\overline{OA}$ and $\overline{OB}$ are two radii of $\odot(O, 35 \text{ cm})$ perpendicular to each other. If $OD = 12 \text{ cm}$,find the area of the shaded region. (in $\text{cm}^2$)