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All courses Geometry · G-C.5 18 of 57
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 \(30^{\circ}\) 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

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

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

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

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

03

Reduce the exact value

\[30\pi/180=(30/180)\pi=\pi/6\]

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

04

Check against a benchmark

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

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.

Solution walkthrough video