How Do You Find the Length of an Arc?

You find the length of an arc using $s=r\theta$, where $r$ is the circle's radius and $\theta$ is the angle at the centre measured in radians. If the angle is given in degrees, convert it to radians first.

Why the Formula Works

A radian is defined using arc length: an angle of one radian subtends an arc whose length equals the radius. Therefore, an angle of $\theta$ radians subtends an arc $\theta$ times as long as the radius:

$$s=r\theta$$

The units of $s$ match the units of $r$. For example, if the radius is measured in centimetres, the arc length is also measured in centimetres.

Another way to see this is as a fraction of the full circumference. A complete circle has angle $2\pi$ and circumference $2\pi r$, so the fraction $\frac{\theta}{2\pi}$ of the circumference gives

$$s=\frac{\theta}{2\pi}(2\pi r)=r\theta.$$

This confirms that the formula applies to any central angle measured in radians.

Use this sequence:

1. Identify the radius and central angle.
2. Convert degrees to radians if necessary.
3. Substitute into $s=r\theta$.
4. State suitable units and round only at the end.

Worked Examples

For a circle with radius $8\text{ cm}$ and central angle $\frac{3\pi}{4}$:

$$s=8\left(\frac{3\pi}{4}\right)=6\pi\text{ cm}$$

The exact arc length is $6\pi\text{ cm}$, or approximately $18.8\text{ cm}$ to three significant figures.

If the radius is $10\text{ m}$ and the angle is $72^\circ$, first convert the angle:

$$72^\circ\times\frac{\pi}{180}=\frac{2\pi}{5}$$

Then:

$$s=10\left(\frac{2\pi}{5}\right)=4\pi\text{ m}$$

If a diagram asks for the major arc, use the reflex central angle, or subtract the minor arc from the full circumference.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches, SL 3.4: Circle, radians, arcs and sectors, and may be tested on Paper 1 or Paper 2. Show the substitution into $s=r\theta$, retain an exact answer involving $\pi$ where appropriate, and otherwise give three significant figures.

A common misconception is using degrees directly in $s=r\theta$. This formula requires radians, so always check the angle unit before substituting.