How to find the arc length of a circle from a diagram

Key takeaway: This kind of arc-length question is really about matching the angle, the radius, and the fraction of the circle. Once those three pieces line up, the rest is quick.

To find the length of arc $PQR$, use the arc-length formula:

$s=\frac{\theta}{360^\circ}\cdot 2\pi r$

or, if the angle is in radians,

$s=r\theta$

What to do

1. Identify the radius of the circle.
2. Identify the central angle that intercepts arc $PQR$.
3. Substitute those values into the correct formula.
4. Leave the answer in terms of $\pi$ if the problem asks for exact form.

If the diagram gives the arc’s central angle in degrees, the answer should stay as a multiple of $\pi$.

A clean final response should look like this:

$\text{arc length}=\text{(number)}\pi$

If you want, I can also show how to read the central angle directly from the figure and avoid the most common setup mistake.

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Pitfalls the pros know 👇 A common slip is using the full circumference formula $2\pi r$ without taking only the circle fraction for the arc. Another easy mistake is mixing degrees and radians in the same formula. If the angle is in degrees, do not plug it into $s=r\theta$ unless you first convert it to radians.

What if the problem changes? If the circle’s central angle changes, the arc length changes proportionally. For example, doubling the central angle doubles the arc length, as long as the radius stays the same. If the arc is a major arc instead of a minor arc, use the larger central angle, not the smaller one.

Tags: arc length, central angle, circumference