A chord $AB$ of a circle, of radius $14 \mathrm{cm}$ makes an angle of $60°$ at the centre of the circle. The area of the minor segment of the circle is $\mathrm{k} {\mathrm{cm}}^2$. Find the value of $\mathrm{k}$.

(Use $\mathrm{\pi }=\frac{22}{7}$)

The diagram shows a sector of a circle of radius $r$ making an angle $\theta °$. The area of the sector is $A {\mathrm{cm}}^2$ and perimeter of the sector is $50 \mathrm{cm}$.

Prove that: $\theta =\frac{360}{\mathrm{\pi }}\left(\frac{25}{r}-1\right)$                       

The diagram shows a sector of a circle of radius $r$ making an angle $\theta °$. The area of the sector is $A {\mathrm{cm}}^2$ and perimeter of the sector is $50 \mathrm{cm}$.

Prove that: $A=25r-r^2$
                                  

The area of the segment $AYB$ (shown in figure), if the radius of the circle is $21 \mathrm{cm}$ and $\angle AOB=120°$, is $\mathrm{k} {\mathrm{cm}}^2$. Find the value of $\mathrm{k}$. (Use $\pi =\frac{22}{7}$)

A chord of a circle subtends an angle of $\theta$ at the centre of the circle. The area of the minor segment cut off by the chord is one-eighth of the area of the circle. Prove that $8 \sin \frac{\theta }{2} \cos \frac{\theta }{2}+\mathrm{\pi }=\frac{\mathrm{\pi }\theta }{45}$.