Radians vs Degrees Explained

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The One Relationship You Need

A half-turn around a circle is \(180^\circ\). That same angle is \(\pi\) radians.

Degrees \(180^\circ\)

=

Radians \(\pi\)

A full turn: \(360^\circ = 2\pi\text{ radians}\)

Converting Angles

There Are Only Two Rules

The conversion factor depends on which unit you are starting with.

1

Degrees → Radians

Multiply by \(\frac{\pi}{180}\).

\[ \boxed{ \text{degrees} \times \frac{\pi}{180} } \]

The degree symbol cancels, leaving radians.

2

Radians → Degrees

Multiply by \(\frac{180}{\pi}\).

\[ \boxed{ \text{radians} \times \frac{180}{\pi} } \]

The \(\pi\) cancels, leaving degrees.

Worked Examples

Converting Degrees to Radians

Multiply the degree measure by \(\frac{\pi}{180}\), then simplify.

Example 1

Convert \(60^\circ\) to radians

Degrees → Radians

Multiply by \(\frac{\pi}{180}\) \[ 60^\circ \times \frac{\pi}{180^\circ} \]

Simplify \[ \frac{60\pi}{180} = \frac{\pi}{3} \]

Answer \[ \boxed{ 60^\circ = \frac{\pi}{3} } \]

Example 2

Convert \(135^\circ\) to radians

Degrees → Radians

Multiply by \(\frac{\pi}{180}\) \[ 135^\circ \times \frac{\pi}{180^\circ} \]

Simplify the fraction \[ \frac{135\pi}{180} = \frac{3\pi}{4} \]

Answer \[ \boxed{ 135^\circ = \frac{3\pi}{4} } \]

Example 3

Convert \(225^\circ\) to radians

Degrees → Radians

Multiply by \(\frac{\pi}{180}\) \[ 225^\circ \times \frac{\pi}{180^\circ} \]

Simplify \[ \frac{225\pi}{180} = \frac{5\pi}{4} \]

Answer \[ \boxed{ 225^\circ = \frac{5\pi}{4} } \]

Worked Examples

Converting Radians to Degrees

Multiply the radian measure by \(\frac{180}{\pi}\), then simplify.

Example 4

Convert \(\frac{\pi}{6}\) to degrees

Radians → Degrees

Multiply by \(\frac{180}{\pi}\) \[ \frac{\pi}{6} \times \frac{180^\circ}{\pi} \]

Cancel \(\pi\) \[ \frac{180^\circ}{6} = 30^\circ \]

Answer \[ \boxed{ \frac{\pi}{6} = 30^\circ } \]

Example 5

Convert \(\frac{5\pi}{4}\) to degrees

Radians → Degrees

Multiply by \(\frac{180}{\pi}\) \[ \frac{5\pi}{4} \times \frac{180^\circ}{\pi} \]

Cancel \(\pi\) and simplify \[ \frac{5(180^\circ)}{4} = 225^\circ \]

Answer \[ \boxed{ \frac{5\pi}{4} = 225^\circ } \]

Example 6

Convert \(\frac{11\pi}{6}\) to degrees

Radians → Degrees

Multiply by \(\frac{180}{\pi}\) \[ \frac{11\pi}{6} \times \frac{180^\circ}{\pi} \]

Cancel \(\pi\) and simplify \[ \frac{11(180^\circ)}{6} = 330^\circ \]

Answer \[ \boxed{ \frac{11\pi}{6} = 330^\circ } \]

Common Angles

Degree and Radian Equivalents

These angle pairs appear constantly in trigonometry. You will eventually want to recognize them without converting each time.

Degrees	Radians	Degrees	Radians
\(0^\circ\)	\(0\)	\(180^\circ\)	\(\pi\)
\(30^\circ\)	\(\frac{\pi}{6}\)	\(210^\circ\)	\(\frac{7\pi}{6}\)
\(45^\circ\)	\(\frac{\pi}{4}\)	\(225^\circ\)	\(\frac{5\pi}{4}\)
\(60^\circ\)	\(\frac{\pi}{3}\)	\(240^\circ\)	\(\frac{4\pi}{3}\)
\(90^\circ\)	\(\frac{\pi}{2}\)	\(270^\circ\)	\(\frac{3\pi}{2}\)
\(120^\circ\)	\(\frac{2\pi}{3}\)	\(300^\circ\)	\(\frac{5\pi}{3}\)
\(135^\circ\)	\(\frac{3\pi}{4}\)	\(315^\circ\)	\(\frac{7\pi}{4}\)
\(150^\circ\)	\(\frac{5\pi}{6}\)	\(330^\circ\)	\(\frac{11\pi}{6}\)
\(180^\circ\)	\(\pi\)	\(360^\circ\)	\(2\pi\)

Why These Angles Matter

These are the same common angles used on the unit circle . Knowing both forms makes later trigonometry much faster.

Going Around the Circle

Coterminal Angles

Coterminal angles end at the same position after rotating around a circle. They differ by one or more complete rotations.

Degrees

Add or subtract \(360^\circ\)

\[ \boxed{ \theta + 360^\circ k } \]

where \(k\) is any integer.

Radians

Add or subtract \(2\pi\)

\[ \boxed{ \theta + 2\pi k } \]

where \(k\) is any integer.

Example 7

Find one positive coterminal angle for \(-45^\circ\)

Degrees

Add one full rotation \[ -45^\circ + 360^\circ \]

Simplify \[ 315^\circ \]

Answer \[ \boxed{ 315^\circ } \]

Example 8

Find one positive coterminal angle for \(-\frac{\pi}{3}\)

Radians

Add one full rotation \[ -\frac{\pi}{3} + 2\pi \]

Write with a common denominator \[ -\frac{\pi}{3} + \frac{6\pi}{3} = \frac{5\pi}{3} \]

Answer \[ \boxed{ \frac{5\pi}{3} } \]

Key Takeaway

You Really Only Need to Remember Three Things

1

Know the anchor

\[ 180^\circ=\pi \]

2

Degrees → Radians

\[ \times \frac{\pi}{180} \]

3

Radians → Degrees

\[ \times \frac{180}{\pi} \]

Check Your Work

Common Radians and Degrees Mistakes

Most mistakes come from using the conversion factor in the wrong direction or forgetting to simplify.

Flipping the Conversion Factor

Degrees to radians uses \(\frac{\pi}{180}\). Radians to degrees uses \(\frac{180}{\pi}\).

Forgetting to Simplify

An answer such as \(\frac{60\pi}{180}\) should be simplified to \(\frac{\pi}{3}\).

Using the Wrong Full Rotation

One full rotation is \(360^\circ\) in degrees and \(2\pi\) in radians.

Treating \(\pi\) Like a Variable

In radian measure, \(\pi\) is part of the exact angle value. Keep it unless you are specifically asked for a decimal approximation.

Ready to Practice?

Try Radian and Degree Conversions

Practice converting both directions, recognizing common angles, and finding coterminal angles.

Practice Problems Unit Circle Practice

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