Integrated Math 2 / Unit 8: Circles
8.5 Arc Length and Sector Area
Notes
A fraction $\displaystyle \frac{\theta}{360}$ of the circle gives the arc and the sector.
- Arc length: $\displaystyle s = \frac{\theta}{360} \cdot 2\pi r$.
- Sector area: $\displaystyle A = \frac{\theta}{360} \cdot \pi r^2$.
- In radians: $s = r\theta$ and $\displaystyle A = \frac{1}{2}r^2\theta$.
- Example: $r = 6$, $\theta = 60°$: $\displaystyle s = \frac{1}{6}(12\pi) = 2\pi$.
Practice
Find the length of a $60°$ arc in a circle of radius 6.
$\displaystyle \frac{60}{360}(2\pi \cdot 6) = \frac{1}{6}(12\pi) = 2\pi$.
Find the area of a $90°$ sector in a circle of radius 10.
$\displaystyle \frac{90}{360}(\pi \cdot 100) = 25\pi$.
An arc of length $5\pi$ lies on a circle of radius 12. What is its central angle?
$\displaystyle 5\pi = \frac{\theta}{360}(24\pi)$, so $\displaystyle \theta = \frac{5 \cdot 360}{24} = 75°$.
Ready for more? Take the Unit 8 test.