Earlier in this module, we used the formula \(\theta = \dfrac{\mbox{arc length}}{\mbox{radius}}\), where \(\theta\) was the angle measured in radians.

By rearranging this formula, we get a formula that allows us to determine the arc length given by the radius and the central angle.

The formula is generally written \(a = r \theta\), where \(\theta\) is the central angle measured in radians, \(r\) is the radius of the circle, and \(a\) is the arc length which subtends \(\theta\).

Determine the arc length which subtends a central angle of \(75^{\circ}\) in a circle whose radius is \(20\) cm.

Solution

Since the angle is given in degrees, it must be converted to radians either separately or inside the formula for arc length.

Converting \(75^{\circ}\) to radians, we get \(\theta = 75^{\circ} \left( \dfrac{\pi}{180^{\circ}}\right) = \dfrac{5 \pi}{12}\).

Substituting \(r = 20\) and \(\theta = \dfrac{5 \pi}{12}\) into \(a = r \theta\), we get \(a = 20 \left(\dfrac{5 \pi}{12} \right )= \dfrac{25 \pi}{3} \approx 26.2\) cm.

Notice that the units for arc length are in centimeters since the units for radius were in centimeters.

The arc length is exactly \(\dfrac{25 \pi}{3}\) cm or approximately \(26.2\) cm.