About Radians and Arc Length
What are Radians and Arc Length?
A radian is a unit of angular measurement defined by the ratio of the arc length to the radius of a circle. Arc length is the distance along the curved edge of a circular arc.
Formulas
The key formulas relating radians, arc length, and radius are:
- Radian measure: \(\theta = \frac{s}{r}\)
- Arc length: \(s = r\theta\)
Where:
- \(\theta\) is the angle in radians
- \(s\) is the arc length
- \(r\) is the radius of the circle
Calculation Steps
- Convert the angle from degrees to radians: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180°}\)
- Calculate the arc length: \(s = r\theta_{rad}\)
- Verify the radian measure: \(\theta_{rad} = \frac{s}{r}\)
Example
Let's calculate the arc length for a circle with radius 5 units and a central angle of 60°:
- Convert 60° to radians: \(\theta_{rad} = 60° \times \frac{\pi}{180°} = \frac{\pi}{3} \approx 1.0472\) radians
- Calculate arc length: \(s = r\theta_{rad} = 5 \times \frac{\pi}{3} \approx 5.2360\) units
- Verify: \(\theta_{rad} = \frac{s}{r} = \frac{5.2360}{5} \approx 1.0472\) radians (matches step 1)