Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter.

The circumference of a circle can be calculated from its diameter using the formula:

$c = π d$

Or, substituting the radius for the diameter:

$c = 2π r$

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...

The circumference of an ellipse is more problematical, the exact solution being an infinite series. A good approximation is Ramanujan's:

$c \approx \pi (3(a+b) - \sqrt{(3a+b)(a+3b)})$

where $a$ and $b$ are the ellipse's semi-major and semi-minor axes, respectively. The two are related by the ellipse's eccentricity as follows:

$b = a \sqrt{1-e^2}$

Which means the circumference can also be written as:

$c \approx \pi a (3(1+\sqrt{1-e^2}) - \sqrt{(3+ \sqrt{1-e^2})(1+3 \sqrt{1-e^2})}) = \pi a (3(1+\sqrt{1-e^2}) - \sqrt{3(2-e^2)+10 \sqrt{1-e^2}})$

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