# Area between Two Non-Intersecting Chords

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## Theorem

Let $AB$ and $CD$ be two chords of a circle whose center is at $O$ and whose radius is $r$.

Let $\alpha$ and $\theta$ be respectively the measures in radians of the angles $\angle COD$ and $\angle AOB$.

Then the area $\AA$ between the two chords is given by:

- $\AA = \dfrac {r^2} 2 \paren {\theta - \sin \theta - \alpha + \sin \alpha}$

if $O$ is not included in the area, and:

- $\AA = r^2 \paren {\pi - \dfrac 1 2 \paren {\theta - \sin \theta + \alpha - \sin \alpha} }$

if $O$ is included in the area.

## Proof

Let $\SS_\alpha$ be the area of the segment whose base subtends $\alpha$.

Let $\SS_\theta$ be the area of the segment whose base subtends $\theta$.

### Case $(1)$: Center included in Area

Let the center $O$ be included in the area.

The area between the two chords is given by:

minus:

Thus:

\(\ds \AA\) | \(=\) | \(\ds \pi r^2 - \SS_\alpha - \SS_\theta\) | Area of Circle: $\pi r^2$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \pi r^2 - \frac 1 2 r^2 \paren {\theta - \sin \theta} - \frac 1 2 r^2 \paren {\alpha - \sin \alpha}\) | Area of Segment of Circle | |||||||||||

\(\ds \) | \(=\) | \(\ds r^2 \paren {\pi - \frac 1 2 \paren {\theta - \sin \theta + \alpha - \sin \alpha} }\) | rearranging |

$\Box$

### Case $(2)$: Center not included in Area

Let $\theta \ge \alpha$.

The area between the two chords is given by:

minus:

Thus:

\(\ds \AA\) | \(=\) | \(\ds \SS_\theta - \SS_\alpha\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 2 r^2 \paren {\theta - \sin \theta} - \frac 1 2 r^2 \paren {\alpha - \sin \alpha}\) | Area of Segment of Circle | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac {r^2} 2 \paren {\theta - \sin \theta - \alpha + \sin \alpha}\) | rearranging |

$\blacksquare$