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Radian to Degree Measure

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Radian to Degree Measure

Study Guide

Key Definition

The measure of an angle in radians is the ratio of the length of the arc to the length of the radius. One full rotation around a circle equals $2\pi$ radians.

Important Notes

One full circle is $360^\circ$ or $2\pi$ radians
To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$
Radians are dimensionless
A straight angle is $\pi$ radians or $180^\circ$
A right angle is $\frac{\pi}{2}$ radians or $90^\circ$

Mathematical Notation

$\pi$ represents the mathematical constant pi

$+$ represents addition

$^\circ$ represents degree measure

$×$ represents multiplication

$/$ represents division

$\frac{a}{b}$ represents a fraction with a horizontal bar

Remember to use proper notation when solving problems

Why It Works

The conversion between radians and degrees is based on the proportionality between a circle's arc length and its radius, using $\frac{180^\circ}{\pi}$ as the conversion factor.

Remember

To convert radians to degrees, use $\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$

Quick Reference

Conversion Formula:$\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$

Understanding Radian to Degree Measure

Choose your learning level

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

At an introductory level, radians measure angles based on the ratio of the arc length to the radius. Since a quarter of a full circle corresponds to an arc equal to the radius, $\frac{\pi}{2} = 90^\circ$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Convert $\frac{\pi}{3}$ radians to degrees.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are designing a sundial. The angle created by the shadow is $\frac{\pi}{4}$ radians. What is this angle in degrees?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explain why the conversion factor $\frac{180^\circ}{\pi}$ is used to convert radians to degrees.

Challenge Quiz

Single Choice Quiz

Advanced

Convert $\frac{7\pi}{6}$ radians to degrees.

Recap

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Review key concepts and takeaways