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

Radian measure expresses an angle using the ratio of arc length to radius. Since $180^\circ=\pi$ radians, multiply degrees by $\frac{\pi}{180}$ to obtain radians and multiply radians by $\frac{180}{\pi}$ to obtain degrees.

Why Radian Measure Works

One radian is the central angle that subtends an arc whose length equals the circle's radius. More generally, if an arc has length $s$, radius $r$, and angle $\theta$ radians, then

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

For a complete circle, the circumference is $2\pi r$. Therefore,

$$\theta=\frac{2\pi r}{r}=2\pi\text{ radians}=360^\circ.$$

Radians are especially useful because they connect angular measure directly to circle geometry. The value of an angle in radians tells you how many radius-lengths fit along its subtended arc. This direct relationship is why the standard arc-length and sector-area formulas use radians.

It follows that $\pi$ radians equals $180^\circ$, giving the two conversion rules:

In each conversion, write the conversion factor explicitly and cancel the original unit. For degree-to-radian conversions, simplify the numerical fraction while keeping $\pi$ exact; use a decimal approximation only when requested.

Worked Conversions

Convert $135^\circ$ to radians:

$$135\times\frac{\pi}{180}=\frac{3\pi}{4}.$$

Convert $\frac{5\pi}{6}$ radians to degrees:

$$\frac{5\pi}{6}\times\frac{180}{\pi}=150^\circ.$$

For example, consider a sector with radius $6\text{ cm}$ and angle $60^\circ$. First, $60^\circ=\frac{\pi}{3}$. Then , and . Converting first prevents degrees being substituted into radian formulas.

A common misconception is that $\pi=180$. This is incorrect: $\pi\text{ radians}=180^\circ$. They represent the same angle in different units.

Radian measure is required in SL 3.4: Circle, radians, arcs, sectors, particularly for $s=r\theta$ and $A=\frac{1}{2}r^2\theta$. Both formulas require in radians.

Exam Technique

IB Mathematics AA assumes radians unless degrees are explicitly stated. On Paper 1, leave answers in exact form involving $\pi$; on Paper 2, check your GDC angle mode and give an exact answer or three significant figures as required.