Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the formula for arc length in degrees?

1. s = rθ

2. s = πrθ/180

3. s = θr²/2

4. s = θπr²/360

The correct answer is: s = πrθ/180

The formula for arc length when the central angle is measured in degrees is derived from the proportional relationship between a circle's circumference and the angle extent of the arc. The total circumference of a circle is given by \(2\pi r\), where \(r\) is the radius of the circle. When the angle is given in degrees, only a fraction of the circle is considered, specifically the fraction that the angle \(θ\) represents out of the full circle (360 degrees). The arc length \(s\) can thus be calculated using this fraction of the circumference, leading to the formula: \[ s = \frac{θ}{360} \times (2\pi r) \] This simplifies to: \[ s = \frac{θ \cdot \pi r}{180} \] This product clearly demonstrates that to find the arc length based on the given angle in degrees, one needs to multiply the angle by the radius and adjust by the factor of \(\pi/180\). Thus, the correct formula for arc length with the central angle in degrees is: \[ s = \frac{\pi r θ}{180} \] This representation keeps all the necessary components—namely, the angle and radius—while