Radian Measure

TipLearning Objectives
  • Convert between degrees and radians.
  • Interpret radian measure as arc-length per radius.
  • Use radian-based formulas for arc length and sector area.

Key Ideas

Definition

A radian is the angle that subtends an arc equal to the radius.

Conversions

\[ 180^\circ = \pi \text{ rad} \]

  • Degrees → radians: multiply by \(\pi/180\)
  • Radians → degrees: multiply by \(180/\pi\)

Radian Arc Formulas

Arc length:
\[ s = r\theta \] Sector area:
\[ A = \frac12 r^2 \theta \]

One radian shown as the angle whose arc length equals the radius.

Common Problem Types

Convert Units

Degrees ↔︎ radians.

Find Arc Length Using \(s = r\theta\)

Only works with radians.

Find Sector Area

Use \(\frac12 r^2\theta\).

Recognizing Exact Radians

\(\pi/6,\pi/4,\pi/3,\pi/2,\pi\).

Strategies

  • Always check units: if using formulas, convert to radians.
  • Keep answers in \(\pi\)-form unless told otherwise.
  • Draw reference circle if needed.

Worked Examples

Example 1

Convert 150° to radians:
\[ 150^\circ \cdot \frac{\pi}{180} = \frac{5\pi}{6}. \]

Example 2

Find arc length if \(r=8\), \(\theta=\frac{\pi}{4}\):
\[ s = r\theta = 8\cdot \frac{\pi}{4} = 2\pi. \]

WarningCommon Mistakes
  • Using degree values directly in \(s=r\theta\).
  • Forgetting to multiply by \(\pi\).
  • Confusing arc-length formulas for radians vs degrees.

Practice Problems

  1. Convert 60° to radians.
  2. Find arc length if \(r=10,\theta=\frac{\pi}{3}\).
  3. Sector area if \(r=6,\theta=\frac{\pi}{2}\).
  1. \(\pi/3\).
  2. \(10 \cdot \frac{\pi}{3} = \frac{10\pi}{3}.\)
  3. \(\frac12 36 \cdot \frac{\pi}{2} = 9\pi.\)

Summary

  • Radians measure arc per radius.
  • Use \(s=r\theta\) and \(\frac12 r^2\theta\).
  • If a formula needs radians, convert first!