Result

Calculation Steps

Visual Representation

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:

1. Radian measure: $\theta =\frac{s}{r}$
2. 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

1. Convert the angle from degrees to radians: ${\theta }_{rad}={\theta }_{deg}\times \frac{\pi }{180°}$
2. Calculate the arc length: $s=r{\theta }_{rad}$
3. 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°:

1. Convert 60° to radians: ${\theta }_{rad}=60°\times \frac{\pi }{180°}=\frac{\pi }{3}\approx 1.0472$ radians
2. Calculate arc length: $s=r{\theta }_{rad}=5\times \frac{\pi }{3}\approx 5.2360$ units
3. Verify: ${\theta }_{rad}=\frac{s}{r}=\frac{5.2360}{5}\approx 1.0472$ radians (matches step 1)