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 |
$\blacksquare$