Modern mathematicians can use more advanced calculus or real analysis of follow-up to get this area of theory. However, the great ancient Greek mathematician Archimedes in the "circle of measurement" (Measurement of a Circle) using Euclidean geometry that a circle is equal to the area within a radius of its length and circumference as the two angle side of the triangle area. Circumference 2π r, right-angled triangle the area of the product of two right-angle side of the half, draw a circle area of π r 2.
Spread the ancient Chinese "Nine Chapters on Arithmetic Katada," chapter in the fields round the area of surgery on the circle described as "half week was multiplied by the radius of the plot step." Wei Liu Hui annotated the times, "Nine Chapters on Arithmetic" when, by the "end" cut round operation that provides the same results. (Source Wikipedia)