Radian

High School

Definition

A way to measure angles, as opposed to using degrees. You can equate radians to degrees as ½π radians equals 90°. This means that 360° is the same as 2π. Radians measure angles using the number of radii required to measure an arc that is described by that angle.

Worked examples

\(\frac{\pi}{2} \) rad\( = 90^\circ\), \(\pi \) rad\( = 180^\circ\), \(2\pi \) rad\( = 360^\circ\)

Common angles: a quarter turn is π/2, a half turn is π, and a full turn is 2π radians.

\(60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) rad\(\)

To convert degrees to radians, multiply by π/180.

\(\frac{3\pi}{4} \) rad\( \times \frac{180^\circ}{\pi} = 135^\circ\)

To convert radians to degrees, multiply by 180/π.

Common mistakes

\(90^\circ = \pi \) rad\(\) → \(90^\circ = \frac{\pi}{2} \) rad\(\) π radians is a half turn (180°), not a quarter turn.
\(\sin(30) = 0.5\) → \(\sin(30^\circ) = 0.5\) or \(\sin\left(\frac{\pi}{6}\right) = 0.5\) Without units, most calculators assume radians; sin(30 rad) ≈ –0.988, not 0.5.
forgetting to leave π in the answer → write \(\frac{\pi}{3}\) instead of 1.047... Exact radian answers are cleaner and more accurate when left in terms of π.

Where you'll use it next

Radians are the standard unit in calculus, where derivatives and integrals of trig functions only work cleanly in radians. You'll also use them in physics for circular motion, waves, and oscillations.

Found in 1 StudyPug lesson

Radian measure and arc length

Grade 12 Math

12th Grade12th