# If the arc length and chord length between two points in a circle are known , find radius of the circle? [duplicate]

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If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle?

## marked as duplicate by hardmath, user98602, Hanul Jeon, Alexander Gruber♦Jun 9 '14 at 5:06

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• What level of geometry do you know? Trigonometric functions? – TZakrevskiy Mar 16 '14 at 11:10

## 1 Answer

If the $O$ is the center of the circle, $A$ and $B$ are the end points of the chord and arc, then et $\alpha$ be an angle $\angle AOB$.

Let also $c$ - the length of the arc and $b$ - the length of $AB$, $d$ - diameter. Then we know that $$\alpha d/2 = c,$$ $$d\sin (\alpha/2)=b,$$ or $$\sin \left(\frac{c}{d}\right)=\frac bd.$$ If $x = \frac cd$, then you are to solve an equation $$\sin x = \frac bc x.$$