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# Relationship Between Arc Length and Arc Measure

The arc length is calculated by multiplying the circle's circumference by the ratio of the measure of the central angle to

Based on the diagram, the following formula is true.

### Proof

Consider the arc in the following diagram.

Since a circle measures this arc represents of Therefore, the ratio of the arc length to the circumference of the whole circle is proportional to
Recall that the circumference of a circle is This expression can be substituted into the equation.
By multiplying both sides of the equation by the desired formula is obtained, which completes the proof.

### Extra

Other Versions of the Formula

From the fact that equals an equivalent formula can be written if the central angle is given in radians.

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