Area between Two Non-Intersecting Chords

Theorem
Let $CD$ and $EF$ be parallel chords of a circle whose center is at $A$ and whose radius is $r$.


 * [[File:Two parallel chords.png]]

Let $\alpha$ and $\theta$ be respectively the measures in radians of the angles $\angle CAD$ and $\angle EAF$, with $\theta > \alpha$.

Then the area $\mathcal A$ between the two chords $EF$ and $CD$ is given by:
 * $\mathcal A = \dfrac {r^2} 2 \left({\theta - \sin \theta - \alpha + \sin \alpha}\right)$, if $A$ is not included in the area


 * $\mathcal A = r^2 \left({\pi - \dfrac 1 2 \left({\theta - \sin \theta + \alpha - \sin \alpha}\right)}\right)$, if $A$ is included in the area.