# 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?

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.

• Thank you for the detailed answer. I was missing the (*) equation. Apr 23 '12 at 13:53
• Actually this answer is incorrect. θ = (d / r), in radians, or θ = (d * 360 / (2 πr)), in degrees. Feb 22 '16 at 3:17
• Unless, of course, you are speaking in turns ;) Feb 24 '16 at 5:55
• @HansBrende: Thank you. I will look at it again, tomorrow perhaps. Feb 24 '16 at 6:05
• @HansBrende: Thank you for spotting the incorrect $\theta$. Fixed. Mar 1 '16 at 6:12

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.

• Thx for the reply, if there would be a distributed solution option I would give the half for u, but since Andrés answer was more detailed... Apr 23 '12 at 13:55