Arc

A circular sector is shaded in green. Its curved boundary of length L is a circular arc.

In 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

$L=\theta r$

An angle of $\alpha$ degrees has a size in radians given by

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

and so the arc length equals

$L=\frac{\pi}{180}\alpha r$

See also

• Arc length
• Other meanings of arc
• Circular-arc graph

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.
• Weisstein, Eric W., "Arc" from MathWorld.

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