9709 · 1.4
Circular measure
Instead of degrees, we can measure angles using 'radians', which are based on the circle's own radius. This makes key formulae for lengths and areas much simpler and more elegant.
Need to know
What you need to know
- To convert degrees to radians, multiply by $\frac{\pi}{180}$.
- To convert radians to degrees, multiply by $\frac{180}{\pi}$.
- Unless a question specifies degrees, assume angles are in radians, especially if $\pi$ is involved.
Explanation
Slicing the Pizza with Pi
- Radian: angle subtended when arc length = radius (θ = s/r). | Sim hint: 360° = 2π rad — convert using π.
- Arc length s = rθ; sector area = ½r²θ (θ in radians). | Sim hint: Use radians in all calculus trig.
- Small angle: sin θ ≈ θ ≈ tan θ for θ in radians. | Sim hint: Check calculator mode — degrees vs radians.
- Angular speed ω = θ/t; v = rω. | Sim hint: Link linear and angular motion.