Rule

Relationship between Arc Length and Arc Measure

In a circle, the ratio between an arc length and the circumference is equal to the ratio between the arc measure and the circle's angle,

36 0^{\circ} .

\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}

Find the arc length

This relationship can be used to calculate the length or measure of an arc.

The arc length between

A

and

B

can be measured using the relationship between the arc length and arc measure. The arc measure is known,

5 0^{\circ},

and with the radius,

4

in., the circumference can be calculated. The circumference is given by the formula

C = 2 π r .

Therefore, the circumference is:

C = 2 π \cdot 4 = 8 π .

The values for the arc measure and the circumference are now used to find the arc length.

\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}

SubstituteValuesSubstitute values

\frac{arc length}{8 π} = \frac{5 0 ^{\circ}}{3 6 0 ^{\circ}}

Solve for

arc length

LHS \cdot 8 π = RHS \cdot 8 π

arc length = \frac{5 0}{3 6 0} \cdot 8 π

CalcQuotProdCalculate quotient and product

arc length = 3.49065 \dots

RoundDecRound to

arc length \approx 3.5

Th arc length can be calculated to

3.5

units.