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2022-04-30

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?

Dexter Conner

Beginner2022-05-01Added 15 answers

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 $\mathrm{\angle}AOB$.

Let also $c$ - the length of the arc and $b$ - the length of $AB$, $d$ - diameter. Then we know that

$\alpha d/2=c,$

$d\mathrm{sin}(\alpha /2)=b,$

or

$\mathrm{sin}\left(\frac{c}{d}\right)=\frac{b}{d}.$

If $x=\frac{c}{d}$, then you are to solve an equation

$\mathrm{sin}x=\frac{b}{c}x.$

Let also $c$ - the length of the arc and $b$ - the length of $AB$, $d$ - diameter. Then we know that

$\alpha d/2=c,$

$d\mathrm{sin}(\alpha /2)=b,$

or

$\mathrm{sin}\left(\frac{c}{d}\right)=\frac{b}{d}.$

If $x=\frac{c}{d}$, then you are to solve an equation

$\mathrm{sin}x=\frac{b}{c}x.$

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