Definitions

# Arc (geometry)

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

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

Substituting in the circumference

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

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

$L=theta r.,!$

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

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

and so the arc length equals then

$L=frac\left\{alphapi r\right\}\left\{180\right\}.,!$