How to calculate the area of a circle ( given: origin, radius ) on a sphere ( Earth )?

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.

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.