# Earth Coverage by Satellites Calculator

An online calculators to calculate the percentage of the area of the earth (assumed to be a sphere) covered by a satellite, given its altitude and angle of elevation, is presented.

## Coverage of Earth by Satellite

In the figure below a satellite at point $S$, in orbit around the earth, can only cover part of the earth that has the shape of spherical cap.

## Ratio $f$ of Surface Area of a Spherical Cap to The Surface of the Earth

The figure below shows a two dimentional representation of a satellite at an altitude $H$ above the surface of the earth. $\theta$ is the angle of coverage and $\gamma$ is the angle of elevation of the satellite , measured with respect to the tangent to the earth.
The area of the spherical cap is given by
$\displaystyle \text{A}_{cap} = 2\pi R h$
The ratio $f$ of the $\displaystyle \text{A}_{cap}$ and the total area of the earth is given by
$f = \dfrac{2\pi R h}{4 \pi R^2} = \dfrac{1}{2} \dfrac{h}{R}$
where $R \approx 6378$ km is the radius of the earth and $h$ is the height of the spherical cap (in red).
$\cos(\alpha) = \dfrac{\overline{OD}}{\overline{OA}} = \dfrac{R-h}{R} = 1 - \dfrac{h}{R}$
hence
$\dfrac{h}{R} = 1 - \cos(\alpha)$
Substitute $\dfrac{h}{R}$ in the formula for $f$ above to obtain
$f = \dfrac{1}{2} (1 - \cos(\alpha)) \quad \quad (I)$
We now need to find $\alpha$
$\alpha + \gamma + \theta + 90^{\circ} = 180^{\circ}$
Use the sine law in triangle $OAS$ to write
$\dfrac{\sin(\gamma+90^{\circ})}{ \overline{OS} } = \dfrac{\sin(\theta)}{ \overline{OA}} \quad \quad (II)$
Note that
$\sin(\gamma+90^{\circ}) = \cos(\gamma)$
$\theta = 90^{\circ}-\alpha - \gamma$
$\sin(\theta) = \sin(90^{\circ}-\alpha - \gamma ) = \cos(\alpha + \gamma)$
$\overline{OS} = H + R$
$\overline{OA} = R$
Substitute in equation (II) and rewrite it as
$\dfrac{\cos(\gamma)}{ H+R } = \dfrac{\cos(\alpha + \gamma)}{ \overline{R}}$
Use cross product to obtain
$\cos(\alpha + \gamma) = \dfrac{R}{H+R} \cos(\gamma)$
Take $arccos$ of both sides
$\alpha + \gamma = \arccos (\dfrac{R}{H+R} \cos(\gamma))$
$\alpha = \arccos (\dfrac{R}{H+R} \cos(\gamma)) - \gamma \quad \quad (III)$

This calculator uses formulae $I$ and $III$ to calculate the percentage of the area of the earth covered by the satellite at an altitude $H$ and angle of elevation $\gamma$

## How to use the calculator

Enter the altitude $H$ of the satellite as a positive real number. Enter the angle of elevation $\gamma$ whose value is in the range $(0,90^{\circ})$ and press "calculate".
The output is the percent of the area of the suface of the earth that is covered by the satellite.

 Altitude of Satellite: $H$ = 36000 Angle of Elevation Gamma: $\gamma$ = 30 degrees Number of Decimals = 3