Circumference
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Circumference is essentially a synonym for perimeter: for a given closed curve in the plane, it is the distance one travels in a complete circuit of the curve. The term circumference is most frequently used to refer to the distance around a circle, though it may refer to the distance around any smooth curve, while the term perimeter is typically reserved for polygons and other non curving shapes.
Formulas
In a circle of radius 
and diameter 
, the circumference 
is given by
![\[C = \pi \cdot d = 2\pi \cdot r\]](http://latex.artofproblemsolving.com/3/5/8/358134d6a09f1e059e89656f96ac8faf3bb4d11a.png)
Indeed, the constant 
(pi) was originally defined to be the ratio of the circumference of a circle to the length of its diameter.
There is, however, no algebraic formula for the circumference of an ellipse (without integrals). Several approximations exist, such as this one:![\[C \approx \pi \left(a + b\right) \left( 1 + \frac{3h}{10 + \sqrt{4 - 3h}} \right) \quad\text{where } h = \frac{\left(a - b\right)^2}{\left(a + b \right)^2}\]](http://latex.artofproblemsolving.com/4/a/b/4ab217396d45db82c06976813a0cfecddcbdb4cc.png)
by Ramanujan.
