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Radians vs Degrees

Overview

The math way to measure.

What is it?

A unit of angle measure based on the radius of the circle.

History

Adopted because it simplifies Calculus formulas. In Calculus, derivatives of trig functions only work cleanly in Radians.

Key Idea

180 degrees = π radians.

Practice This Topic

Concept Guide

Plain English: Degrees (360) are arbitrary numbers from ancient Babylonians. Radians are physical. One Radian is the angle you get when you wrap the radius along the edge of the circle.

Real-world example: Car tires. When calculating how fast a car moves based on wheel rotation (angular velocity), you must use Radians.

How to do it

  1. To get Radians: Multiply Degrees by (π / 180).
  2. To get Degrees: Multiply Radians by (180 / π).
  3. Simplify the fraction (don't type π into the calculator unless you need a decimal).
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Common Pitfall

Multiplying by the wrong conversion factor. Remember: You want to cancel out the unit you started with.

Word Problem
"You eat a slice of pizza that is 45 degrees. How much is that in Radians?"
Reasoning: Take 45. Multiply by π/180. 45/180 simplifies to 1/4. The answer is π/4 radians.

Practice Examples

Conversion
Deg -> Rad: * π/180 Rad -> Deg: * 180/π
Think: 'Put the unit I want on top'.