Circumference
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.
Circle
The circumference of a circle can be calculated from its diameter using the formula:
<math>c = \pi d<math>
Or, substituting the radius for the diameter:
<math>c = 2 \pi r<math>
Where r is the radius and d is the diameter of the circle, and π (the Greek letter pi) is the constant 3.141 592 6...
Ellipse
The circumference of an ellipse is more problematical, the exact solution being an infinite series. A good approximation is Ramanujan's:
<math>c \approx \pi (3(a+b) – \sqrt{(3a+b)(a+3b)})<math>
where <math>a<math> and <math>b<math> are the ellipse's semi-major and semi-minor axes, respectively. The two are related by the ellipse's eccentricity as follows:
<math>b = a \sqrt{1-e^2}<math>
Which means the circumference can also be written as:
<math>c \approx \pi a (3(1+\sqrt{1-e^2}) – \sqrt{(3+ \sqrt{1-e^2})(1+3 \sqrt{1-e^2})}) = \pi a (3(1+\sqrt{1-e^2}) – \sqrt{3(2-e^2)+10 \sqrt{1-e^2}})<math>