# Arc Length of Sector

Jump to navigation
Jump to search

## 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:

## Proof

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

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

This article, or a section of it, needs explaining.In particular: Why is the density of the arc length uniform? i.e. why does equal rotation sweeps out equal arc length?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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: $4.14$