Area of a circle each polygon side is a radius, of length "r". And since the total side length is greater than the circumference, the polygon consists of "n" identical triangles with total area greater than "T". Again we have a contradiction, so our supposition that "C" might be less than "T" must be wrong as well. Therefore, it must be the case that the area enclosed by the circle is precisely the same as the area of the triangle. This concludes the proof. Following Satō Moshun and Leonardo da Vinci , we can use inscribed regular polygons in a different way. Suppose we
Area of a circle good approximation to the circumference. To compute "u" and "U" for large "n", Archimedes derived the following doubling formulae: Starting from a hexagon, Archimedes doubled "n" four times to get a 96-gon, which gave him a good approximation to the circumference of the circle. In modern notation, we can reproduce his computation (and go further) as follows. For a unit circle, an inscribed hexagon has "u" = 6, and a circumscribed hexagon has "U" = 4. Doubling seven times yields The last entry of the table has ⁄ as one of its best rational approximations; i.e., there is no better