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
Circumference in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse, is Some lower and upper bounds on the circumference of the canonical ellipse with formula_5 are Here the upper bound formula_9 is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound formula_10 is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes. The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind. More precisely, we