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
Earth's circumference shadow on the ground. Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. This angle was about 7°, or 1/50th the circumference of a circle; taking the Earth as perfectly spherical, he concluded that the Earth's circumference was 50 times the known distance from Alexandra to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia. Depending on whether he used the "Olympic stade" (176.4m) or the Italian stade (184.8m), this would imply a circumference of 44,100 km (an error of