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Based on the diagram, the following formula is true.

$s=360_{∘}θ ⋅2πr$

Consider the arc $s$ in the following diagram.

Since a circle measures $360_{∘},$ this arc represents $360_{∘}θ $ of $⊙C.$ Therefore, the ratio of the arc length $s$ to the circumference of the whole circle is proportional to $360_{∘}θ .$$Circumferences =360_{∘}θ $

Recall that the circumference of a circle is $2πr.$ This expression can be substituted into the equation. $2πrs =360_{∘}θ $

By multiplying both sides of the equation by $2πr,$ the desired formula is obtained, which completes the proof. $s=360_{∘}θ ⋅2πr$

From the fact that $360_{∘}$ equals $2πrad,$ an equivalent formula can be written if the central angle is given in radians.

$s=2πθ ⋅2πr⇓s=θr $

Since the measure of an arc is equal to the measure of its central angle, the arc $AB$ measures $θ.$ Therefore, by substituting $mAB$ for $θ,$ another version of the formula is obtained that can also be written in degrees or radians.

$s=360_{∘}mAB ⋅2πrors=mAB⋅r $