An arc's measure from Central Angles and Arcs is in degrees. Its length is a distance along the circle, in units like centimeters. To get the length, take the same fraction of the whole circumference.
Why it works
A central angle of $\theta$ degrees cuts off $\frac{\theta}{360}$ of the circle, so its arc is $\frac{\theta}{360}$ of the circumference $2\pi r$. Because all circles are similar, the ratio $\frac{s}{r}$ for a given angle is the same in every circle. That ratio is the angle's radian measure: the arc length measured in radii. A full turn is $\frac{2\pi r}{r} = 2\pi$ radians, so $180^\circ = \pi$ radians.
Key idea
$$s = \frac{\theta}{360} \cdot 2\pi r \quad (\theta \text{ in degrees})$$ $$s = r\theta \quad (\theta \text{ in radians})$$ $$180^\circ = \pi \text{ radians}$$
Worked example: In degrees
Find the length of a $135^\circ$ arc of a circle with radius $8$.
- Fraction of the circle: $\frac{135}{360} = \frac{3}{8}$. Circumference: $2\pi \cdot 8 = 16\pi$.
- $s = \frac{3}{8} \cdot 16\pi = 6\pi \approx 18.85$.
Answer: $6\pi$
Worked example: Given the diameter
A circle has diameter $30$ cm. How long is an arc with a $48^\circ$ central angle?
- Radius: $15$ cm, so the circumference is $30\pi$.
- $s = \frac{48}{360} \cdot 30\pi = \frac{2}{15} \cdot 30\pi = 4\pi \approx 12.6$ cm.
Answer: $4\pi$ cm
Worked example: Radians
Convert $150^\circ$ to radians, then find the arc it cuts off in a circle of radius $6$.
- $150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}$ radians.
- $s = r\theta = 6 \cdot \frac{5\pi}{6} = 5\pi$.
Answer: $\frac{5\pi}{6}$ radians; $s = 5\pi$
Radians make arc length simple: $s = r\theta$. An angle of $2$ radians in a circle of radius $5$ cuts off an arc of length $10$. To convert, multiply degrees by $\frac{\pi}{180}$: $48^\circ = \frac{48\pi}{180} = \frac{4\pi}{15}$ radians. One radian is about $57.3^\circ$, a little less than the $60^\circ$ angle of an equilateral triangle.
Common mistake
Using the diameter as $r$, or forgetting to take the fraction. With $d = 30$, the radius is $15$; and a $48^\circ$ arc is only $\frac{48}{360}$ of the circumference, not all of it. Check: the arc must be shorter than the whole circumference.