Radian Measure Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radian Measure.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An angle measure defined by the arc length subtended on a unit circle: one radian is the angle that subtends an arc equal in length to the radius.

Imagine wrapping the radius of a circle along its edge like a piece of string. The angle you've swept out is exactly 1 radian. Since the full circumference is 2\pi r, a full turn is 2\pi radians. Radians measure angles in terms of the circle itself, which is why they're the natural unit for calculus and physicsβ€”no arbitrary conversion factor like 360 is needed.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Radians connect angle measure directly to arc length: on a circle of radius r, an angle of \theta radians subtends an arc of length s = r\theta.

Common stuck point: Students often forget to switch their calculator to radian mode. If you get unexpected trig values, check your mode first.

Sense of Study hint: Memorize the one key fact: 180 degrees = pi radians. To convert, multiply degrees by pi/180 or radians by 180/pi.

Worked Examples

Example 1

easy
Convert 135Β° to radians and \dfrac{5\pi}{6} radians to degrees. Show the conversion steps.

Solution

  1. 1
    135Β° to radians: multiply by \frac{\pi}{180}. 135\times\frac{\pi}{180}=\frac{135\pi}{180}=\frac{3\pi}{4} rad.
  2. 2
    \frac{5\pi}{6} rad to degrees: multiply by \frac{180}{\pi}. \frac{5\pi}{6}\times\frac{180}{\pi}=\frac{5\times180}{6}=\frac{900}{6}=150Β°.
  3. 3
    Memory aid: \pi rad = 180Β°; to go to radians multiply by \pi/180; to go to degrees multiply by 180/\pi.

Answer

135Β° = \dfrac{3\pi}{4} rad; \dfrac{5\pi}{6} rad = 150Β°
Radians and degrees both measure angles; \pi radians = 180Β° is the fundamental conversion. Radians are preferred in calculus because they make derivative formulas for trig functions clean (no extra factors).

Example 2

medium
A wheel of radius 5 cm rotates through an angle of 2.4 radians. Find the arc length and the area of the sector swept.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Convert: (a) 60Β° to radians, (b) 270Β° to radians, (c) \dfrac{\pi}{3} to degrees, (d) \dfrac{7\pi}{6} to degrees.

Example 2

medium
A clock's minute hand is 15 cm long. How far does its tip travel in 20 minutes? How large is the sector area swept?
← Back to Radian Measure Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

unit circlepi