Integrated Math 3 / Unit 5: Trigonometric Functions
5.6 Arc Length and Sector Area
Notes
For a circle of radius $r$ and a central angle $\theta$ in radians:
- Arc length $s = r\theta$.
- Sector area $\displaystyle A = \frac{1}{2}r^2\theta$.
- Example: $r = 6$ and $\displaystyle \theta = \frac{\pi}{3}$ give $\displaystyle s = 6 \cdot \frac{\pi}{3} = 2\pi$.
- If the angle is in degrees, convert to radians first.
Practice
A circle has radius 6 and a central angle of $\displaystyle \frac{\pi}{3}$ radians. What is the arc length?
$\displaystyle s = r\theta = 6 \cdot \frac{\pi}{3} = 2\pi$.
What is the area of a sector with radius 4 and central angle $\displaystyle \frac{\pi}{2}$?
$\displaystyle A = \frac{1}{2}(16)\left(\frac{\pi}{2}\right) = 4\pi$.
An arc of length 12 cm lies on a circle of radius 8 cm. What is the central angle?
$\displaystyle \theta = \frac{s}{r} = \frac{12}{8} = 1.5$.
Ready for more? Take the Unit 5 test.