# 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:

[itex]c = \pi d[itex]

Or, substituting the radius for the diameter:

[itex]c = 2 \pi r[itex]

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:

[itex]c \approx \pi (3(a+b) - \sqrt{(3a+b)(a+3b)})[itex]

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

[itex]b = a \sqrt{1-e^2}[itex]

Which means the circumference can also be written as:

[itex]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}})[itex]

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