Sector & Arc Length — Formulas & Examples

A sector is the region between two radii and the arc joining them. Arc length is the length of that curved edge; sector area is the area of the slice. Both depend on the radius $r$, the central angle $\theta$, and one detail that decides which formula you use: whether $\theta$ is in radians or degrees.

When to use which formula

The very first check is the angle unit, because it splits the formulas into two cases.

If $\theta$ is in radians:

If $\theta$ is in degrees:

The radian formulas only work when the angle is actually in radians, which is the condition behind the most common mistake on this topic. The two cases are really the same idea: a sector takes a fraction of a full circle. A full circle has circumference $2\pi r$ and area $\pi r^2$, and the central angle sets the fraction. A $90^\circ$ angle is one quarter of a turn, so its sector has a quarter of the circumference and a quarter of the area. In radians the fraction is just $\frac{\theta}{2\pi}$, so $\theta = \pi/3$ gives $\frac{1}{6}$ of the circle.

The procedure, step by step

1. Read the radius and angle. Note $r$ and whether the central angle is in degrees or radians.
2. Find the arc length. Use $s = r\theta$ for radians or $s = \frac{\theta}{360} \cdot 2\pi r$ for degrees.
3. Find the sector area. Use $A = \frac{1}{2}r^2\theta$ for radians or $A = \frac{\theta}{360} \cdot \pi r^2$ for degrees.
4. Keep units straight. Arc length uses length units; sector area uses square units.

A fast way to pick the right formula is to ask two questions before computing: do I need the curved distance or the inside area, and is the angle in degrees or radians?

Full example: radius 12 cm, angle 60 degrees

A sector has radius $12$ cm and central angle $60^\circ$. The angle is in degrees, so use the degree formulas.

Arc length:

so the arc length is $4\pi$ cm. Sector area:

so the area is $24\pi\ \text{cm}^2$. There is a built-in check, since for any sector

Using $r = 12$ and $s = 4\pi$,

which matches, confirming the setup is consistent.

Where students get stuck, and how to self-check

"Is my angle in the right unit?" Using $s = r\theta$ with $\theta$ still in degrees is the classic error. Confirm the unit before applying a formula.

"Radius or diameter?" The formulas need the radius. Halve the diameter if that is what the problem gives.

"Arc or chord?" Arc length follows the curve; a chord is a straight segment. They are not interchangeable.

"Right units on the answer?" Sector area must come out in square units, not length units.

"Did I round too early?" If the problem wants an exact answer in terms of $\pi$, keep $\pi$ symbolic instead of rounding mid-calculation.

Practice this yourself

Take radius $9$ m and central angle $120^\circ$. Find the arc length first, then the sector area, and confirm that $A = \frac{1}{2}rs$ gives the same area. That cross-check tells you whether both the formulas and the units hold together.

Where these formulas are used

Sector area and arc length appear in geometry and trigonometry whenever you work with part of a circle: wheels, gears, circular tracks, pie-chart slices, and engineering drawings. They also carry into physics and calculus, where radians make rotation formulas simpler and more consistent.