NASA has built a service for estimating populations, part of which is a map hosted by Columbia on which you can draw a circle or a shape and the map will give you an estimate of the population contained by the shape. You can also access the service via an API…just send it polygonal coordinates and it returns population data.

Just for fun, I drew a small circle with an area of ~7000 sq. km and dragged it around to different spots on the globe:

NYC: 15,251,980

Rural Wyoming: 31

London: 12,972,590

Lagos, Nigeria: 14,106,980

Mumbai: 24,513,630

Tibet: 2,226

Seoul: 23,439,930

Tokyo: 32,572,740

Mexico City: 21,319,990