Degrees and Radians - Precalculus & Trigonometry Step by Step

Convert angles between degree measure and exact π-fraction radian measure using the relationship π rad = 180°.

Key conversions: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π. Example: 135° × π/180 = 135π/180 = 3π/4 rad. Example: 5π/6 × 180/π = 5·180/6 = 150°.

Example

Convert between degrees and exact symbolic radian measure.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

\begin{array}{l}\text{Convert } 135 ^\circ\text{ to radians}\\ \text{Convert } \frac{5\pi}{6} \text{ to degrees}\end{array}

Step 2 — Use radians factor

Convert degrees to radians by multiplying 135 by pi over 180.

\hl{135} ^\circ\cdot\frac{\pi}{ \hl{180} }

Step 3 — Simplify radians

Simplify 135pi over 180 to 3pi over 4.

135 \cdot\frac{\pi}{ 180 }= \hlmath{\frac{3\pi}{4}}

Step 4 — Use degree factor

Convert radians to degrees by multiplying 5pi over 6 by 180 over pi.

\hlmath{\frac{5\pi}{6}} \cdot\frac{ \hl{180} }{\pi}

Step 5 — Simplify degrees

Cancel pi and simplify: 5 times 180 divided by 6 is 150 degrees.

5 \cdot 180 / 6 = \hl{150} ^\circ

degrees-radians Two common angle units: Degrees → Radians: multiply by π/180, then reduce the fraction. Radians → Degrees: multiply by 180/π, then reduce.