How to Convert Degrees and Radians

The formulas for converting between degrees and radians are: $$ \alpha ° = \alpha_{rad} \cdot \frac{180°}{ \pi } $$ $$ \alpha_{rad} = \alpha ° \cdot \frac{ \pi }{ 180° } $$

A Practical Example

Consider an angle measuring 60°

$$ \alpha = 60° $$

The equivalent measure in radians is \( \frac{1}{3} \pi \)

$$ \alpha_{rad} = \alpha ° \cdot \frac{ \pi }{ 180° } $$

$$ \alpha_{rad} = 60° \cdot \frac{ \pi }{ 180° } $$

$$ \alpha_{rad} = \frac{ 1 }{ 3 } \pi $$

Example 2

Consider an angle measuring \( \frac{2}{3} \pi \) radians

$$ \alpha = \frac{2}{3} \pi \ rad $$

The equivalent measure in degrees is 120°

$$ \alpha ° = \alpha_{rad} \cdot \frac{180°}{ \pi } $$

$$ \alpha ° = \frac{2}{3} \pi \cdot \frac{180°}{ \pi } $$

$$ \alpha ° = \frac{2}{3} \cdot 180° $$

$$ \alpha ° = 120° $$

The Proof

To demonstrate the conversion formulas between degrees and radians, note that an angle in degrees is to an angle in radians as a full circle (360°) is to 2π radians.

$$ \alpha ° \ : \ \alpha_{rad} = 360° \ : \ 2 \pi $$

Therefore,

$$ \frac{ \alpha ° }{ \alpha_{rad} } = \frac{ 360° }{ 2 \pi } $$

Simplifying,

$$ \frac{ \alpha ° }{ \alpha_{rad} } = \frac{ 180° }{ \pi } $$

By isolating the angle in degrees, we obtain the first formula:

$$ \alpha ° = \alpha_{rad} \cdot \frac{ 180° }{ \pi } $$

By isolating the angle in radians, we obtain the second formula:

$$ \alpha_{rad} = \alpha ° \cdot \frac{ \pi }{ 180° } $$

And that's how it's done.