In a circle with radius $21 \ cm$,the perimeter of a minor sector is $64 \ cm$. Then,the length of the arc of that sector is $\ldots \ cm$.

In a circle with radius $20 \,cm$,the measures of the angle subtended at the centre for two distinct sectors are $15^{\circ}$ and $90^{\circ}$. Then,the ratio of the areas of those sectors is $\ldots \ldots \ldots$.

In $\odot(O, r)$,the length of minor $\widehat{ACB}$ is $\frac{1}{6}$ times the circumference of the circle. Then,the measure of the angle subtended at the centre by minor $\widehat{ACB}$ is ......... (in $^{\circ}$)

The central angles of two sectors of circles of radii $7 \, cm$ and $21 \, cm$ are respectively $120^{\circ}$ and $40^{\circ}$. Find the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?

On a square cardboard sheet of area $784 \, cm^{2}$, four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates (in $cm^{2}$).

As shown in the diagram,the radii of two concentric circles are $21 \, cm$ and $28 \, cm$. If $m \angle AOB = 40^\circ$,find the area of the shaded region. (in $cm^2$)