Circular measure
Radians
- A radian is the angle that cuts off an arc equal to the radius.
- The link: $\pi \text{ radians} = 180^\circ$.
- degrees → radians: multiply by $\dfrac{\pi}{180}$; radians → degrees: multiply by $\dfrac{180}{\pi}$.
Practice
π radians equals how many degrees?
By definition, π radians = 180°.
Arc length and sector area
- For a sector with radius $r$ and angle $\theta$ in radians:
- arc length $s = r\theta$,
- sector area $A = \tfrac12 r^2 \theta$.
- Segment area = sector − triangle = $\tfrac12 r^2(\theta - \sin\theta)$.
Practice
A sector has radius 5 and angle 2 radians. What is the arc length (s = rθ)?
s = rθ = 5 × 2 = 10.
Practice
A sector has radius 4 and angle 0.5 radians. What is its area (A = ½r²θ)?
A = ½ × 4² × 0.5 = ½ × 16 × 0.5 = 4.
You've got it
Key idea
- $\pi$ radians $= 180^\circ$; convert by $\times\dfrac{\pi}{180}$ or $\times\dfrac{180}{\pi}$
- arc length $s = r\theta$; sector area $A = \tfrac12 r^2\theta$ (θ in radians)
- segment $= \tfrac12 r^2(\theta - \sin\theta)$