10.6 Arc Length and Sector Area
Notes
Arc length and sector area are fractions of the whole circle, using the central angle $\theta$.
- Arc length: $\displaystyle L = \frac{\theta}{360^\circ} \cdot 2\pi r$.
- Sector area: $\displaystyle A = \frac{\theta}{360^\circ} \cdot \pi r^2$.
- In radians, $\displaystyle \theta = \frac{\text{arc length}}{r}$, and $180^\circ = \pi$ radians. Then $L = r\theta$.
- Example: $r = 6$, $\theta = 60^\circ$. $\displaystyle L = \frac{1}{6} \cdot 12\pi = 2\pi$. $\displaystyle A = \frac{1}{6} \cdot 36\pi = 6\pi$.
Practice
Find the length of an arc with a central angle of $90^\circ$ in a circle of radius 8.
$\displaystyle \frac{90}{360} \cdot 16\pi = 4\pi$.
Find the area of a sector with a central angle of $120^\circ$ and radius 9.
$\displaystyle \frac{120}{360} \cdot 81\pi = 27\pi$.
Convert $135^\circ$ to radians.
$\displaystyle 135 \cdot \frac{\pi}{180} = \frac{3\pi}{4}$.
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