Key takeaway: A radian is the angle at a circle’s center that opens onto a rim stretch as long as the radius, so a half turn of 180 degrees is pi radians.
Grade 10 – GeometryG-C.B.5Trigonometry
Definition of Radian
One radian is what you get by bending a piece of string as long as the radius around the rim of a circle: the angle that string subtends at the center, about \(57.3^{\circ}\).
The whole rim holds \(2\pi\) such strings, so one full turn is \(2\pi\) of this unit and a half turn is \(\pi\). Mathematicians prefer it because rim distance becomes the plain product \(s = r\theta\). If a trig or calculus formula shows an angle without any unit, this one is normally meant.
Formula and notation
Converting a radian to degrees and back follows from \(180^{\circ} = \pi\) rad:
\[ \theta_{\text{rad}} = \theta_{\text{deg}}\times \frac{\pi}{180}, \qquad \theta_{\text{deg}} = \theta_{\text{rad}}\times \frac{180}{\pi} \]
For \(\theta\) in radians, the rim distance is \(s = r\theta\).
Key facts
- One of these units is about 57.3 degrees.
- A full turn is 2 pi and a quarter turn is pi over 2.
- Inside trig and calculus formulas, no unit label is written.
- Arc length is simply radius times angle when the angle uses this unit.
Radian example
Convert \(150^{\circ}\) to radians. Then convert \(2.4\) rad to degrees, and find the rim distance for \(2.4\) rad on a wheel of radius 7 cm.
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Degrees to rad: \(150\times \frac{\pi}{180} = \frac{5\pi}{6}\).
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Rad to degrees: \(2.4\times \frac{180}{\pi}\approx 137.5^{\circ}\).
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Rim distance: \(s = r\theta = 7\times 2.4 = 16.8\) cm.
\(150^{\circ} = \) \(\frac{5\pi}{6}\) rad, \(2.4\) rad \(\approx\) \(137.5^{\circ}\), and the rim distance is 16.8 cm.
Common mistake
Watch out
Dario converts \(150^{\circ}\) by multiplying by \(\frac{180}{\pi}\) and gets about 8,594.
Why it is wrong
He used the factor meant for converting back to degrees. Going from degrees to radians needs \(\frac{\pi}{180}\). A quick check: 150 degrees is less than a half turn, so the answer must be less than \(\pi \approx 3.14\).
Related terms and lessons
- Geometry · Arc Length, Sector Area and RadiansLessonWorksheetQuiz
- Algebra 2 · Radians and the Unit CircleLessonWorksheetQuiz
- Precalculus · Angles, Radians and the Unit CircleLessonWorksheetQuiz
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