Length of Arc of Small Circle

Theorem
Let $S$ be a sphere.

Let $\bigcirc FCD$ be a small circle on $S$.

Let $C$ and $D$ be the points on $\bigcirc FCD$ such that $CD$ is the arc of $\bigcirc FCD$ whose length is to be determined.

Construction
Let $P$ and $Q$ be the poles of $\bigcirc FCD$.

Let $\bigcirc PCQ$ and $\bigcirc PDQ$ be great circles on $S$.

Let $\bigcirc EAB$ be the great circle whose poles are $P$ and $Q$.

Let $A$ and $B$ be the points on $\bigcirc EAB$ which intersect $\bigcirc PCQ$ and $\bigcirc PDQ$.

The length of the arc $CD$ of $\bigcirc FCD$ is given by:
 * $CD = AB \cos AC$

or:
 * $CD = AB \sin PC$

Proof

 * Size-of-small-circle.png

Let $R$ denote the center of $\bigcirc FCD$.

Let $O$ denote the center of $S$, which is also the center of $\bigcirc EAB$.

We have:
 * $CD = RC \times \angle CRD$

Similarly:
 * $AB = OA \times \angle AOB$

By Circles with Same Poles are Parallel:
 * $\bigcirc FCD \parallel \bigcirc EAB$

Hence $RC$ and $RD$ are parallel to $OA$ and $OB$ respectively.

Thus:

We also have that:

and that:

We have that $\angle AOC$ is the (plane) angle subtended at $O$ by the arc $AC$ of $\bigcirc EAB$.

Thus:

Hence the result.