Arc Length of Sector

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Theorem

Let $\mathcal C = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$.

Let $BAC$ be the sector of $\mathcal C$ whose angle between $AB$ and $AC$ is $\theta$.

Sector.png


Then the length $s$ of arc $BC$ is given by:

$\mathcal s = r \theta$

where:

$r = AB$ is the length of the radius of the circle
$\theta$ is measured in radians.


Proof

From Perimeter of Circle, the perimeter of $\mathcal C$ is $2 \pi r$.

From Full Angle measures $2 \pi$ Radians, the angle within $\mathcal C$ is $2 \pi$.



The fraction of the perimeter of $\mathcal C$ within the sector $BAC$ is therefore $2 \pi r \times \dfrac \theta {2 \pi}$.

Hence the result.

$\blacksquare$


Sources