Trigonometry / Unit 1: Angles and Their Measure
1.4 Arc Length and Sector Area
Notes
On a circle of radius $r$, a central angle $\theta$ cuts off an arc of length $s$ and a sector of area $A$. The angle must be in radians.
- Arc length: $s = r\theta$.
- Sector area: $\displaystyle A = \frac{1}{2}r^2\theta$.
- If the angle is in degrees, convert it first.
- Example: $r = 4$, $\displaystyle \theta = \frac{\pi}{2}$ gives $\displaystyle s = 4 \cdot \frac{\pi}{2} = 2\pi$ and $\displaystyle A = \frac{1}{2}(16)\frac{\pi}{2} = 4\pi$.
Practice
A circle has radius 6 and a central angle of $\displaystyle \frac{\pi}{3}$ radians. What is the arc length?
$\displaystyle s = r\theta = 6 \cdot \frac{\pi}{3} = 2\pi$.
A central angle of $120^\circ$ is drawn in a circle of radius 9. What is the arc length?
$\displaystyle 120^\circ = \frac{2\pi}{3}$ rad, so $\displaystyle s = 9 \cdot \frac{2\pi}{3} = 6\pi$.
A sector has radius 10 and central angle $\displaystyle \frac{\pi}{5}$ radians. What is its area?
$\displaystyle A = \frac{1}{2}(10)^2 \cdot \frac{\pi}{5} = 50 \cdot \frac{\pi}{5} = 10\pi$.
Ready for more? Take the Unit 1 test.