Squaring the circle the circular arc b around O with radius , it results S'. Bisect the line segment in D and draw the semicircle b over D. Draw a straight line from O through C up to the semicircle b, it cuts b in E. The line segment is the mean proportional between and , also called geometric mean. Extend the line segment beyond O and transfer twice more, it results F and A, and thus the length of the line segment with the above described approximation value of formula_3, the half circumference of the circle. Bisect the line segment in G
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