Arc_(geometry)

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

In Euclidean geometry, a circular arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, an arc is a segment of the circumference. 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 $\backslash theta\backslash ,\backslash !$ (measured in radians) with the circle center—i.e., the central angle—equals $\backslash theta\; r\backslash ,\backslash !$. This is because

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

Substituting in the circumference

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

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

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

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

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

and so the arc length equals then

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

See also

• Arc length
• Other meanings of arc

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.

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