In geometry you found arc length and sector area as a fraction of the whole circle, with the angle in degrees. Radians make those formulas much shorter, because a radian is already measured in radii.
Key idea
With $\theta$ in radians: $$s = r\theta \qquad A = \tfrac{1}{2}r^2\theta.$$ For turning motion, angular speed is $\omega = \frac{\theta}{t}$ and the speed of a point on the rim is $v = r\omega$.
Why it works
A sector of angle $\theta$ is the fraction $\frac{\theta}{2\pi}$ of the full circle. So its arc is $\frac{\theta}{2\pi} \cdot 2\pi r = r\theta$ and its area is $\frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta$. For speed: in time $t$ a point on the rim travels the arc $s = r\theta$, so its speed is $\frac{s}{t} = r \cdot \frac{\theta}{t} = r\omega$.
Worked examples
Worked example: Arc and area
A sector has radius $6$ in and angle $\frac{2\pi}{3}$. Find its arc length and area.
- Arc: $s = r\theta = 6 \cdot \frac{2\pi}{3} = 4\pi$ in.
- Area: $A = \frac{1}{2} \cdot 6^2 \cdot \frac{2\pi}{3} = 18 \cdot \frac{2\pi}{3} = 12\pi$ square inches.
Answer: $s = 4\pi \approx 12.6$ in, $A = 12\pi \approx 37.7$ in²
Worked example: Given in degrees
Find the arc length cut off by a $45^\circ$ angle on a circle of radius $8$ cm.
- Convert first: $45^\circ = \frac{\pi}{4}$.
- $s = 8 \cdot \frac{\pi}{4} = 2\pi$ cm.
Answer: $2\pi \approx 6.28$ cm
Worked example: A spinning wheel
A bike wheel of radius $30$ cm turns at $100$ revolutions per minute. How fast is the bike moving?
- Each revolution is $2\pi$ radians, so $\omega = 100 \cdot 2\pi = 200\pi$ radians per minute.
- $v = r\omega = 30 \cdot 200\pi = 6000\pi$ cm per minute.
- That is about $18{,}850$ cm, or $188.5$ m, each minute.
Answer: $6000\pi$ cm/min, about $11.3$ km/h
Common mistake
Using degrees in $s = r\theta$. With $45^\circ$ and $r = 8$ you would get $s = 360$ cm, but the whole circle is only $2\pi \cdot 8 \approx 50$ cm around. Check: an arc can't be longer than the circumference unless the angle is more than a full turn.