# 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

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:

$AB = OA \times \angle AOB$
$\bigcirc FCD \parallel \bigcirc EAB$

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

Thus:

 $\ds \angle CRD$ $=$ $\ds \angle AOB$ $\ds \leadsto \ \$ $\ds CD$ $=$ $\ds \dfrac {RC} {OA} AB$ $\ds$ $=$ $\ds \dfrac {RC} {OC} AB$ as $OA = OC$ are both radii of $S$

We also have that:

 $\ds RC$ $\perp$ $\ds OR$ $\ds \leadsto \ \$ $\ds RC$ $=$ $\ds OC \cos \angle RCO$

and that:

 $\ds RC$ $\parallel$ $\ds OA$ $\ds \leadsto \ \$ $\ds \angle RCO$ $=$ $\ds \angle AOC$

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

Thus:

 $\ds CD$ $=$ $\ds AB \cos AC$ $\ds$ $=$ $\ds AB \, \map \cos {PA - PC}$ $\ds$ $=$ $\ds AB \sin PC$ as $PA$ is a right angle, and Cosine of Complement equals Sine‎

Hence the result.

$\blacksquare$