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A circular sector is shaded in green with length L along the circle's perimeter

A circular sector is shaded in green with length L along the circle's perimeter

In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle. If the arc segment occupies a great circle (or great ellipse), it is considered a great-arc segment.

The length of an arc of a circle with radius $r$ and subtending an angle $\theta\,\!$ (measured in radians) with the circle center—i.e., the central angle—equals $\theta r\,\!$ . This is because

$\frac{L}{\mathrm{circumference}}=\frac{\theta}{2\pi}.\,\!$

Substituting in the circumference

$\frac{L}{2\pi r}=\frac{\theta}{2\pi},\,\!$

and solving for arc length, $L$ , in terms of $\theta\,\!$ yields

$L=\theta r.\,\!$

For an angle $α$ measured in degrees, the size in radians is given by

$\theta=\frac{\alpha}{180}\pi,\,\!$

and so the arc length equals then

$L=\frac{\alpha\pi r}{180}.\,\!$

See also

External links

Definition and properties of a circular arc With interactive animation
A collection of pages defining arcs and their properties, with animated applets Arcs, arc central angle, arc peripheral angle, central angle theorem and others.
Eric W. Weisstein, Arc at MathWorld.

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