九章算術
When , half the circumference of the -gon approaches a semicircle, thus half a circumference of a circle multiplied by its radius equals the area of the circle. Liu Hui did not explain in detail this deduction. However it is self-evident by using Liu Hui's "in-out complement principle" which he provided elsewhere in "The Nine Chapters on the Mathematical Art": Cut up a geometric shape into parts, rearrange the parts to form another shape, the area of the two shapes will be identical.