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The arc length is calculated by multiplying the circle's circumference by the ratio of the measure of the central angle to 360^(∘).
Based on the diagram, the following formula is true.
s = θ/360^(∘) * 2π r
Since a circle measures 360^(∘), this arc represents θ360^(∘) of ⊙ C. Therefore, the ratio of the arc length s to the circumference of the whole circle is proportional to θ360^(∘). s/Circumference = θ/360^(∘) Recall that the circumference of a circle is 2π r. This expression can be substituted into the equation. s/2π r = θ/360^(∘) By multiplying both sides of the equation by 2π r, the desired formula is obtained, which completes the proof.
s = θ/360^(∘) * 2π r
s = θ/2π * 2π r [0.6em] ⇓ s = θ r
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 = mAB/360^(∘) * 2π r [0.6em] or [0.4em] s = mAB* r