Area of a circle Archimedes is not rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident. The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes' argument in "The
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