How to calculate the area of a circle ( given: origin, radius ) on a sphere ( Earth )? I know that the Earth isn't a sphere, not even an ellipsoid, but for my measurements, its an acceptable approximation.
Assuming I have a coordinate(lat,lon) and a distance( e.g.: 1000km ), what is the surface area on the earth with that distance radius?
 A: Just a small amplification of the answer by Ross Millikan. Use the same notation as the article he linked to.  I take it that your $1000$ km is the surface of the Earth distance from the center $C$ of your circle to the furthest points $P$ from the center. In the picture linked to, $C$ is the top of the sphere, and $P$ is any point on outer edge of the bottom of the cap.
Assume that this surface of the Earth distance is $d$, and that is is $\le$ $1/4$ of the circumference of the Earth (that's not necessary, but it makes visualization easier).  Let the radius of the Earth be $r$.
Then the angle $\theta$ subtended by the arc $CP$ at the centre of the Earth is given by
$$\theta=\frac{d}{r}.\tag{$\ast$}$$
The "$h$" in the linked picture is given by 
$h=r-r\cos\theta$. The surface area is $2\pi rh$, which is
$$2\pi r^2(1-\cos\theta).\tag{$\ast\ast$}$$ 
Compute $\theta$ using $(\ast)$, and then use $(\ast\ast)$ to find the surface area.
A: You could look at Spherical Cap for the formula.  If you treat the earth as a sphere, the  coordinates of the center do not matter, just the radius and the radius of the earth.
