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, and sector area is the area of the slice.

If a circle has radius $r$ and central angle $\theta$, first check the angle unit. If $\theta$ is in radians, use

$$s = r\theta$$

and

$$A = \frac{1}{2}r^2\theta$$

If $\theta$ is in degrees, use

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

and

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

The condition matters. The radian formulas work only when the angle is measured in radians.

Why the formulas work

Both formulas come from taking a fraction of a whole circle.

A full circle has circumference $2\pi r$ and area $\pi r^2$. A sector takes only the fraction set by the central angle. For example, $90^\circ$ is one quarter of a full turn, so its sector has one quarter of the circle's circumference and one quarter of its area.

In radians, the same idea becomes cleaner because a full circle is $2\pi$ radians. If the angle is $\pi/3$, the sector is $\frac{\pi/3}{2\pi} = \frac{1}{6}$ of the circle.

That is why both quantities grow in a predictable way: a larger radius makes both larger, and a larger central angle also makes both larger.

Worked example: radius $12$ cm, angle $60^\circ$

Suppose a sector has radius $12$ cm and central angle $60^\circ$.

Because the angle is in degrees, use the degree formulas.

For arc length,

$$s = \frac{60}{360} \cdot 2\pi(12)$$ $$s = \frac{1}{6} \cdot 24\pi = 4\pi$$

So the arc length is $4\pi$ cm.

For sector area,

$$A = \frac{60}{360} \cdot \pi(12)^2$$ $$A = \frac{1}{6} \cdot 144\pi = 24\pi$$

So the sector area is $24\pi\ \text{cm}^2$.

There is a useful check here. For the same sector,

$$A = \frac{1}{2}rs$$

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

$$A = \frac{1}{2}(12)(4\pi) = 24\pi$$

The result matches, so the setup is consistent.

Common mistakes with sector area and arc length

1. Using $s = r\theta$ when $\theta$ is still in degrees.
2. Using the diameter where the formulas need the radius.
3. Mixing up arc length and chord length. Arc length follows the curve; a chord is a straight segment.
4. Forgetting that sector area must be written in square units.
5. Rounding too early when the problem wants an exact answer in terms of $\pi$.

When sector area and arc length are used

These formulas show up in geometry and trigonometry whenever you are working with part of a circle instead of the whole circle. Common examples include wheels, gears, circular tracks, slices of pie charts, and engineering drawings.

They also matter later in physics and calculus because radians make rotation formulas simpler and more consistent.

Quick way to choose the right formula

Ask two questions first:

1. Do I need the curved distance or the inside area?
2. Is the angle in degrees or radians?

If you answer those correctly, the right formula is usually obvious.

Try a similar problem

Try your own version with radius $9$ m and central angle $120^\circ$. Find the arc length first, then the sector area, and check whether $A = \frac{1}{2}rs$ gives the same area. That is a good next step if you want to test whether the formulas and units both make sense.