Area between Two Non-Intersecting Chords

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


 * Circle with chords and area.png
 * Circle with chords and area 3.1.png

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

Then the area $\mathcal A$ between the two chords is given by:
 * $\mathcal A = \dfrac {r^2} 2 \left({\theta - \sin \theta - \alpha + \sin \alpha}\right)$

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


 * $\mathcal A = r^2 \left({\pi - \dfrac 1 2 \left({\theta - \sin \theta + \alpha - \sin \alpha}\right)}\right)$

if $O$ is included in the area.

Proof
Let $\mathcal S_\alpha$ be the area of the segment whose base subtends $\alpha$.

Let $\mathcal S_\theta$ 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:
 * the area of the whole circle

minus:
 * the areas of the segments $\mathcal S_\alpha$ and $\mathcal S_\theta$.

Thus:

Case $(2)$: Center not included in Area
Let $\theta \ge \alpha$.

The area between the two chords is given by:
 * the area of the segment $\mathcal S_\theta$

minus:
 * the area of the segment $\mathcal S_\alpha$.

Thus: