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All courses Math II · G-C.5 33 of 73
Derive arc length and sector area formulas using similarity; define radians and convert degrees/radians.

Convert an angle from degrees to radians by multiplying by π/180

Problem
Convert 30 degrees to radians.
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Hint

Multiply the degree measure by \(\pi/180\) to convert to radians.

\(180^\circ = \pi\) radians, so degrees convert to radians by multiplying by \(\pi/180\).

Solution walkthrough

01

Choose a unit-canceling conversion factor

\[\begin{aligned} 180~\text{degrees}=\text{pi}~\text{radians} \\ \text{conversion}~\text{factor}=\text{pi}~\text{radians}/(180~\text{degrees}) \end{aligned}\]

The angle starts in degrees and must end in radians, so place degrees in the denominator. The factor equals 1 because pi radians and 180 degrees name the same angle.

02

Convert before simplifying

\[30~\text{degrees}~*~(\text{pi}~\text{radians})/(180~\text{degrees})=30\text{pi}/180~\text{radians}\]

The degree units cancel, leaving radians. This makes the conversion direction explicit rather than relying on memorization alone.

03

Reduce the exact value

\[30\text{pi}/180=(30/180)\text{pi}=\text{pi}/6\]

Divide numerator and denominator by 30. The exact radian measure is pi/6.

04

Check against a benchmark

\[\begin{aligned} 30~\text{degrees}=(1/6)(180~\text{degrees}) \\ \text{therefore}~\text{radian}~\text{measure}=(1/6)\text{pi} \end{aligned}\]

Thirty degrees is one-sixth of a straight angle, so pi/6 radians has the correct size and units.

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Another way

  1. Use the benchmark 60 degrees=pi/3 radians and take half of both measures.

!

Common mistake

Do not multiply by 180/pi. That factor places radians in the denominator and converts radians to degrees, the opposite direction.