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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.


In a circle of radius $r$ and diameter $d = 2r$, the circumference $C$ is given by \[C = \pi \cdot d = 2\pi \cdot r\] Indeed, the constant $\pi$ (pi) was originally defined to be the ratio of the circumference of a circle to the length of its diameter.

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