Length of Chord of Circle

From ProofWiki
Jump to: navigation, search

Theorem

Let $C$ be a circle of radius $r$.

Let $AB$ be a chord which joins the endpoints of the arc $ADB$.

Then:

$AB = 2 r \sin \dfrac \theta 2$

where $\theta$ is the angle subtended by $AB$ at the center of $C$.


Proof

LengthOfChord.png

Let $O$ be the center of $C$.

Let $AB$ be bisected at $E$ by $OD$.

By Conditions for Diameter to be Perpendicular Bisector, $\angle OEA$ is a right angle.

By definition of sine:

$\sin \dfrac \theta 2 = AE / AO$

But $AB = 2 AE$ and $AO = r$.

Hence:

$AB = 2 r \sin \dfrac \theta 2$

$\blacksquare$


Historical Note

This result was the basis of the calculations that Hipparchus of Nicaea used when creating his trigonometrical tables.


Sources