Angles & Angle Measure

Q: Which way do I multiply?

Degrees to radians: times $\dfrac{\pi}{180}$. Radians to degrees: times $\dfrac{180}{\pi}$. Set it up so the old unit cancels.

Trigonometry measures angles two ways: degrees and radians. Radians tie angle measure to the circle itself, and $\pi$ radians equals $180^\circ$ .

Converting between them is a single multiplication by the right form of $\dfrac{\pi}{180}$ or $\dfrac{180}{\pi}$ . Getting the factor's direction right is the whole trick.

The conversion factor

Since $\pi$ radians $= 180^\circ$ , multiply degrees by $\dfrac{\pi}{180}$ to get radians, or multiply radians by $\dfrac{180}{\pi}$ to get degrees.

Set it up so the unit you are leaving cancels. Degrees times $\dfrac{\pi}{180}$ leaves radians.

Simplify the result

After multiplying, reduce the fraction. $150^\circ$ times $\dfrac{\pi}{180}$ becomes $\dfrac{150\pi}{180} = \dfrac{5\pi}{6}$ .

Radian answers are usually left in terms of $\pi$ as an exact value.

Worked examples

Example 1: degrees to radians

Convert $150^\circ$ to radians.

Multiply by pi over 180

150 \cdot \dfrac{\pi}{180}

Simplify the fraction

\dfrac{150\pi}{180}

Reduce

\dfrac{5\pi}{6}

Answer: $\dfrac{5\pi}{6}$

Example 2: radians to degrees

Convert $\dfrac{\pi}{3}$ to degrees.

Multiply by 180 over pi

\dfrac{\pi}{3} \cdot \dfrac{180}{\pi}

Simplify

60^\circ

Answer: $60^\circ$

Try one yourself

Common questions

How many radians is 180 degrees?

Exactly $\pi$ radians. That equality is the basis of every degree-radian conversion.

Which way do I multiply?

Degrees to radians: times $\dfrac{\pi}{180}$ . Radians to degrees: times $\dfrac{180}{\pi}$ . Set it up so the old unit cancels.

Should I leave pi in the answer?

For radian measures, yes — keep the exact form like $\dfrac{5\pi}{6}$ unless a decimal is requested.

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Angles & Angle Measure

The conversion factor

Simplify the result

Worked examples

Example 1: degrees to radians

Example 2: radians to degrees

Try one yourself

Common questions

Keep learning

Trigonometric Functions of General Angles

Graphing Sine & Cosine Functions

Law of Sines & Law of Cosines