Arc Length of Sector

Theorem

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

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


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

$s = r \theta$

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 $\CC$ is $2 \pi r$.

From Measurement of Full Angle, the angle within $\CC$ is $2 \pi$.


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

Hence the result.

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Sector of Circle of Radius $r$: $4.14$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Sector of Circle of Radius $r$: $7.14.$
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