Radians Examples in Math

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

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

Concept Recap

A radian measures angle by arc length: one radian subtends an arc equal to the circle radius.

It ties angle directly to the circle’s geometry instead of degree counting.

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 are natural angle units for trigonometry and calculus.

Common stuck point: Most calculus formulas (derivatives of trig functions, arc length) are only correct when angles are in radians β€” using degrees silently breaks the formulas.

Sense of Study hint: Convert units first, then evaluate trig expressions.

Worked Examples

Example 1

easy
Convert 150Β° to radians.

Solution

  1. 1
    Use the conversion factor: 180Β° = \pi radians.
  2. 2
    Multiply: 150Β° \times \frac{\pi}{180Β°} = \frac{150\pi}{180}.
  3. 3
    Simplify: \frac{150\pi}{180} = \frac{5\pi}{6}.

Answer

\frac{5\pi}{6} \text{ radians}
Radians measure angles by the arc length subtended on a unit circle. Since the full circle has circumference 2\pi, a full rotation is 2\pi radians = 360Β°. The conversion factor \frac{\pi}{180} converts degrees to radians.

Example 2

medium
Find the arc length of a sector with radius 10 cm and central angle \frac{3\pi}{4} radians.

Practice Problems

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

Example 1

medium
Convert \frac{7\pi}{4} radians to degrees and identify which quadrant this angle is in.

Example 2

hard
A wheel of radius 30 cm rotates at 120 rpm (revolutions per minute). Find the angular velocity in radians per second and the linear speed of a point on the rim.
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Background Knowledge

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

piarc lengthunit circle