Finding the area of regions other than polygons proved to be much more difficult during the times of the ancient Greeks.
These Greeks were able to determine formulas for the areas of some general regions by the exhaustion method. The clearest use of such a method was shown by Archimedes who used it to derive formulas for the areas of ellipses, parabolic segments, and sectors of a spiral.
The exhaustion method is a limiting process in which the area of the region is squeezed between two polygons. One polygon is inscribed in the region and the other is circumscribed about it.
When attempting to find the area of a circle, this method can be used. For example, the area of a circular region is approximated by an n-sided inscribed polygon and an n-sided circumscribed polygon. For each value of n, the area of the inscribed polygon is less than the area of the circle, and the area of the circumscribed polygon is greater than the area of the circle. Keeping that in mind, as n increases, the areas of both polygons become better and better approximations of the area of the circle.
A method similar to this one, involving riemann sums, is used in approximating the area of a plane region, or the area underneath a curve.
!! Horrible ASCII
ahead! (Use your imagination
/ x x \
/ x ------ \
/ / \ x \
/ x / \ \
\ \ / x /
\ x \ / /
\ ------ x /
\ x x /
The x's represent the circle. The slash
es and dash
es represent the polygon. In this case, it is a hexagon
As you can see from the beautiful ASCII above, one polygon is inscribed inside the circle, and the other is circumscribed about it. Therfore, the area of the circle must be some area between the area of the bigger polygon minus the area of the smaller polygon. As the sides on the polygons increase, the difference between the two areas of the polygons decrease. Hence the approximation of the area of the circle becomes better.