A satellite is orbiting around the earth in an orbit in equatorial plane, of radius $2R_e$ , where $R_e$is the radius of the earth. Find the area on the earth, this satellite covers for communication purpose in its complete revolution?

$\sin \theta =\frac{\mathrm{R}}{2\mathrm{R}}=\frac{1}{2};\theta ={30}^{\circ }\Rightarrow \alpha ={30}^{\circ }$

For solid angle

Unconvered area $=2\pi R^2(1-\cos \alpha )$

$=2\pi {\mathrm{R}}^2(1-\frac{\sqrt{3}}{2})=\pi {\mathrm{R}}^2(2-\sqrt{3})$

$\therefore$ Total unconvered area  $={\mathrm{U}}_{\mathrm{a}}$

${\mathrm{U}}_{\mathrm{a}}=2\pi {\mathrm{R}}^2(2-\sqrt{3})$  Total surface area of the

$\text{Earth}=4\pi R^2$

$\therefore$ Coverage area on the Earth in one complete rotation is $\mathrm{A}=4\pi {\mathrm{R}}^2-{\mathrm{U}}_{\mathrm{a}}$

$A=4\pi R^2-2\pi R^2[2-\sqrt{3}];A=2\pi R^2\sqrt{3}$