Archimedes's Method for Determining the Area of a Circle
This Demonstration shows how Archimedes determined the area of a circle.
Almost 23 centuries ago, Archimedes determined the area of a circle in terms of , its circumference. This was a remarkable feat, especially since this was an exact result, relying only on logical arguments and not involving the calculation of the numerical value of . A detailed description of the problem and Archimedes's reasoning appears in Dunham .
The result states that the area of a circle of radius is equal to the area of a right-angle triangle, with one side of length and the other the circumference of the circle.
Two basic facts are used to establish the result.
1. The area of a regular polygon with apothem and perimeter is . (An apothem of a regular polygon is a line segment or the distance from the center to the midpoint of a side.)
2. The essence of the method of exhaustion: given a preassigned area, no matter how small, one can find an inscribed (or circumscribed) regular polygon for which the difference between the circle's area and the polygon's is less than this preassigned amount .
 W. Dunham, Journey through Genius: The Great Theorems of Mathematics, New York: Penguin Books, 1991, pp. 90–92.