The diagram shows a semicircle with radius $10 \mathrm{cm}$ and centre $O$. Angle $BOC=\theta$ radians. The perimeter of sector $AOC$ is twice the perimeter of sector $BOC$. Find the perimeter of the triangle$ABC$.

The diagram shows the cross-section of two cylindrical metal rods of radii$x \mathrm{cm}$ and $y \mathrm{cm}$. A thin band, of length $P \mathrm{cm}$, holds the two rods tightly together. Show that $P=4\sqrt{xy}+\pi (x+y)+2(x-y){\sin }^{-1}\left(\frac{x-y}{x+y}\right)$.

Find, in terms of $\mathrm{\pi }$ the area of a sector of: radius $12 \mathrm{cm}$ and angle $\frac{\pi }{6}$ radians is of the form of $n\mathrm{\pi }$, then $n=$

Find, in terms of $\mathrm{\pi }$ the area of a sector of radius $10 \mathrm{cm}$ and angle $\frac{2\pi }{5}$ radians.

Find, in terms of $\mathrm{\pi }$ the area of a sector (in ${\mathrm{cm}}^2$) of radius $4.5 \mathrm{cm}$ and angle $\frac{2\pi }{9}$ radians is of the form of $n\mathrm{\pi }$, then $n=$

Find, in terms of $\mathrm{\pi }$ the area of a sector of radius $9 \mathrm{cm}$ and angle $\frac{4\pi }{3}$ radians in ${\mathrm{cm}}^2$ is of the form of $n\mathrm{\pi }$, then $n=$

Find the area of a sector of radius $34 \mathrm{cm}$ and angle $1.5$ radians.

Find the area of a sector of radius $2.6 \mathrm{cm}$ and angle $0.9$ radians.[Answer up to three decimals] 

Find, in radians, the area of a sector of radius $4 \mathrm{cm}$ and area $9 {\mathrm{cm}}^2$.