Radian
In short: A radian is the size of the central angle whose arc is exactly as long as the circle’s radius, so a full turn measures 2π.
GeometryTrigonometryG-C.B.5
What is Radian?
Bend a string as long as a circle’s radius along the rim, and the central opening it spans is one radian, about \(57.3^{\circ}\). The circumference holds \(2\pi\) such strings, so a full turn equals \(2\pi\) of these units.
That gives the conversion \(180^{\circ} = \pi\). To change degrees into this unit, multiply by \(\frac{\pi}{180}\); to go back, multiply by \(\frac{180}{\pi}\). The unit is handy because arc length becomes simply \(s = r\theta\), with no extra constants.
How to spot it
- An angle is written with \(\pi\), such as \(\frac{3\pi}{4}\), and carries no degree symbol.
- A problem says the angle is in radians, or warns you to switch the calculator mode.
- Arc length or sector area is requested with a short formula that avoids fractions of \(360\).
- A unit circle or sine wave is labeled with multiples of \(\frac{\pi}{6}\), \(\frac{\pi}{4}\) or \(\frac{\pi}{2}\).
The method
- To convert from degrees, multiply by \(\frac{\pi}{180}\) and simplify the fraction.
- To convert to degrees, multiply by \(\frac{180}{\pi}\); the \(\pi\) cancels.
- Remember the anchors: \(\pi\) is \(180^{\circ}\), \(\frac{\pi}{2}\) is \(90^{\circ}\), \(2\pi\) is \(360^{\circ}\).
- For arc length, multiply the radius by the angle measured this way: \(s = r\theta\).
Radian: solved example
Problem
Convert \(210^{\circ}\) to radians and \(\frac{7\pi}{4}\) to degrees.
Solution
From degrees: \(210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}\).
To degrees: \(\frac{7\pi}{4} \times \frac{180}{\pi} = \frac{7 \times 180}{4} = 315\). The conversions are \(\frac{7\pi}{6}\) and \(315^{\circ}\).
Try it
Question 1
Rewrite \(40^{\circ}\) with \(\pi\) in simplest form.
Show the answer
\(40 \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9}\).
Question 2
A pendulum swings through \(1.5\) radians and its string is 8 inches long. How long is the path of its tip?
Show the answer
\(s = r\theta = 8 \times 1.5 = 12\) inches.
Radian vs. Central Angle
A central angle is the shape: two radii opening from the center. The radian is one way of measuring that opening, just as inches and centimeters are two ways of measuring a length. The same opening can be written \(90^{\circ}\) or \(\frac{\pi}{2}\).
Keep going
Lessons
Worksheets and quizzes
- Radians and the Unit Circle – Worksheet
- Radians and the Unit Circle – Quiz
- Arc Length and Sector Area – Worksheet
- Arc Length and Sector Area – Quiz
Related words
Whole dictionary: A–Z · Geometry · Trigonometry