p(k)=k*(49/80);

c1=0.0000122, c2=-0.0021973, c3=0.919315, c4=-1.0359227, c5=0.861913

c6=-0.7274398, c7=0.6352295, c8=-0.436051, c9=0.1818904, c10=-0.0333691

This is one of the most accurate formulas not requiring a very lengthy series expansion or similar polynomial expression for the circumference of an ellipse over the range from the degenerate collapsed ellipse to circle. This formula is also more direct than many as it avoids the use of the Hoelder mean. The maximum relative error is less than 50 ppb (max error at the aspect ratio of about 1:0.00024), permitting 7 significant digits.

For a highly accurate ellipse circumference calculator permitting 8 significant digits, and without the use of π, click below:

# Highly Accurate Ellipse Circumference Calculator

Febraury 25, 2012 by Thomas Blankenhorn