Calculate Satellite Gain: Earth-to-Sphere Ratio

[bos1234]

This is not a connection. I just want ot know how they dervied this formula from gain

Consider a geosynchronous satellite whose global beam covers all of the visible Earth's surface. The radius of the Earth is about 6400km, and the altitude of the satellite is 36,000km.

How would you work out the gain of the satellite?

Book has :Satellite antenna gain, relative to the isotropic antenna situation, is equivalent to the inverse ratio of the cross-secitonal area of the Earth to the surface area of a sphere at 36,000km, assuming 100% effeciency.

Solution has $$\frac{4pi R^2}{pi R^2}$$

numerator is for satellite and denominator is for earth

The Attempt at a Solution

 

[tiny-tim]

hi bos1234!

Book has :Satellite antenna gain, relative to the isotropic antenna situation, is equivalent to the inverse ratio of the cross-secitonal area of the Earth to the surface area of a sphere at 36,000km, assuming 100% effeciency.

the cross-sectional area of the Earth isn't the area of the whole sphere, it's only the area of a slice through the centre of the sphere, so it's only πR² instead of 4πR²