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Rule

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

36 0^{\circ} .

Based on the diagram, the following formula is true.

$s = \frac{θ}{3 6 0 ^{\circ}} \cdot 2 π r$

Proof

Consider the arc $s$ in the following diagram.

Since a circle measures

36 0^{\circ},

this arc represents

\frac{θ}{3 6 0 ^{\circ}}

⊙ C .

Therefore, the ratio of the arc length

s

to the circumference of the whole circle is proportional to

\frac{θ}{3 6 0 ^{\circ}} .

\frac{s}{Circumference} = \frac{θ}{3 6 0 ^{\circ}}

Recall that the circumference of a circle is

2 π r .

This expression can be substituted into the equation.

\frac{s}{2 π r} = \frac{θ}{3 6 0 ^{\circ}}

By multiplying both sides of the equation by

2 π r,

the desired formula is obtained, which completes the proof.

$s = \frac{θ}{3 6 0 ^{\circ}} \cdot 2 π r$

Extra

Other Versions of the Formula

From the fact that $36 0^{\circ}$ equals $2 π rad,$ an equivalent formula can be written if the central angle is given in radians.

s = \frac{θ}{2 π} \cdot 2 π r ⇓ s = θ r

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

s = \frac{m A B}{3 6 0 ^{\circ}} \cdot 2 π r or s = m A B \cdot r

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Proof

Extra