Degrees and Radians: How to Convert and Why It Matters

To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$. A full turn is $360^\circ = 2\pi$ radians, so $180^\circ = \pi$. Radians are the natural unit in calculus because they make the derivative of $\sin x$ exactly $\cos x$.

What is a radian?

Take a circle of radius $r$. Measure an arc along the edge that is also $r$ long. The angle at the centre is one radian, about $57.3^\circ$.

Because the whole circumference is $2\pi r$, a full turn is $2\pi$ radians.

Degrees to radians

Example: $135^\circ$

Simplify the fraction $\frac{135}{180}$ by dividing by 45.

Radians to degrees

Example: $\frac{5\pi}{6}$

Example without π: $1$ radian $= \frac{180}{\pi} \approx 57.2958^\circ$.

Common angles

Why radians?

• Arc length and sector area are simpler: arc $= r\theta$ and area $= \frac{1}{2}r^2\theta$, with $\theta$ in radians.
• Calculus works cleanly: $\frac{d}{dx}\sin x = \cos x$ only when $x$ is in radians. In degrees an extra factor of $\frac{\pi}{180}$ appears.
• Small angles: for small $\theta$ in radians, $\sin\theta \approx \theta$.

Common mistakes

• Using the wrong calculator mode. $\sin 30$ in radian mode is about $-0.988$, not $0.5$.
• Multiplying by $\frac{180}{\pi}$ when going the other way.
• Leaving fractions unsimplified, such as $\frac{90\pi}{180}$ instead of $\frac{\pi}{2}$.

Practice questions

1. Convert $210^\circ$ to radians.
2. Convert $\frac{7\pi}{4}$ to degrees.
3. Convert $2.5$ radians to degrees (1 decimal place).

Answers: 1) $\frac{7\pi}{6}$ 2) $315^\circ$ 3) $143.2^\circ$

Use the degrees to radians calculator for exact π answers and the radians to degrees calculator for the reverse. Then find exact trig values with the trigonometry calculator, or read sine, cosine and tangent explained.

Frequently asked questions

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