Arc Length Formula

The arc length formula gives the distance along part of a circle. If a circle has radius $r$ and central angle $\theta$ in radians, then

$$s = r\theta$$

If the angle is given in degrees instead, use

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

Both formulas say the same thing: the arc length is the same fraction of the circumference as the central angle is of a full turn.

What arc length means

Arc length is not the straight-line distance between two points. It is the length you would measure if you traced the curve itself.

In a circle, two things control that length. The radius tells you how large the circle is, and the central angle tells you how much of the circle you are taking.

A larger radius gives a longer arc. A larger angle also gives a longer arc.

Why $s = r\theta$ only works with radians

Radians are defined using arc length. One radian is the angle that cuts off an arc equal in length to the radius, so when $\theta = 1$, the formula gives $s = r$.

That is why the radian formula is so clean. A full circle has angle $2\pi$ radians and circumference $2\pi r$, so taking a fraction $\frac{\theta}{2\pi}$ of the circle gives

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

If the angle is in degrees, convert first or use the degree formula. The condition matters: $s = r\theta$ is correct only when $\theta$ is in radians.

Worked example with a degree angle

Suppose a circle has radius $10$ m and central angle $72^\circ$. Since the angle is in degrees, use

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

Substitute $\theta = 72$ and $r = 10$:

$$s = \frac{72}{360} \cdot 2\pi(10)$$

Now simplify:

$$s = \frac{1}{5} \cdot 20\pi = 4\pi$$

So the exact arc length is $4\pi$ m.

If you want a decimal approximation,

$$4\pi \approx 12.57$$

so the arc length is about $12.57$ m.

You can also convert $72^\circ$ to radians:

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

Then

$$s = r\theta = 10 \cdot \frac{2\pi}{5} = 4\pi$$

Both methods agree, which is a good check.

Common arc length mistakes

1. Using $s = r\theta$ when the angle is still in degrees.
2. Using the diameter where the formula needs the radius.
3. Confusing arc length with chord length. Arc length follows the curve, while a chord is the straight segment between the same endpoints.
4. Mixing arc length with sector area. Sector area uses a different formula.

When the arc length formula is used

The circle version appears in geometry, trigonometry, and applied problems with wheels, gears, circular tracks, and rotation.

In calculus, the idea extends to general curves. If $y = f(x)$ is smooth enough on an interval $[a,b]$, the arc length is

$$L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx$$

That formula is for the length of a graph, not just part of a circle. The condition matters here too: the derivative must exist on the interval, and the integral must make sense.

Quick check before you finish

If the angle doubles and the radius stays the same, the arc length doubles.

If the radius doubles and the angle stays the same, the arc length also doubles.

If your answer does not scale that way, recheck the angle unit and whether you used radius or diameter.

Try a similar problem

Try your own version with radius $6$ cm and central angle $150^\circ$. Solve it once with the degree formula and once by converting to radians first. If both answers match, your setup is solid.