What Is a Radian and How Do You Convert Between Degrees and Radians?

A radian is the angle at the centre of a circle subtended by an arc whose length equals the circle's radius. Convert degrees to radians by multiplying by $\frac{\pi}{180}$, and convert radians to degrees by multiplying by $\frac{180}{\pi}$.

Why the Conversion Works

For an angle $\theta$ measured in radians,

$$\theta=\frac{s}{r},$$

where $s$ is the arc length and $r$ is the radius. Therefore, when $s=r$, the angle is one radian.

The circumference of a circle is $2\pi r$, so one complete revolution contains

$$\frac{2\pi r}{r}=2\pi$$

radians. Since a complete revolution is also $360^{\circ}$,

$$2\pi\text{ radians}=360^{\circ},$$

and therefore

$$\pi\text{ radians}=180^{\circ}.$$

This equivalence produces both conversion rules.

Two further examples show how to preserve exact values and use decimal approximations correctly. For $225^{\circ}$,

$$225^{\circ}\times\frac{\pi}{180}=\frac{5\pi}{4}.$$

Dividing the numerator and denominator by $45$ gives the exact simplified result. For $1.2$ radians,

$$1.2\times\frac{180}{\pi}=68.754\ldots^{\circ}\approx68.8^{\circ}.$$

Keep the full calculator value until the final rounding step. Radians also simplify circle formulas: when $\theta$ is in radians, arc length is $s=r\theta$ and sector area is $A=\frac{1}{2}r^2\theta$. Do not use these forms with an angle measured in degrees.

Exact radian answers are normally written using $\pi$. For example, $60^{\circ}=\frac{\pi}{3}$, rather than the decimal approximation $$ radians.

A common misconception is reversing the conversion factors. Check whether the result is reasonable: converting degrees to radians usually makes the numerical value smaller, while converting radians to degrees usually makes it larger.

IB Exam Technique

This is AHL 3.7 Radian measure, so it is HL-only content and can be tested on any GDC-active Math AI HL paper. On HL examinations, assume angles are in radians unless a degree symbol is shown. Keep exact multiples of $\pi$ where possible, otherwise give the final answer to three significant figures, and always check that the GDC is in the required angle mode.