Radians: why 180° = π, and what a radian actually is

Degrees are familiar and they subdivide neatly — $1^\circ = 60'$ and $1' = 60''$ — but radians are the measure mathematics actually prefers. Students usually meet them as a conversion rule to be memorised, which hides how simple the idea is.

The definition

Take a circle of radius $R$. Its circumference is $2R\pi$. Now cut off an arc whose length is exactly $R$ — the same as the radius. The central angle standing on that arc is one radian.

That is the whole definition. A radian is not a number pulled from nowhere; it is the angle you get when the arc and the radius are the same length.

Where 180° = π comes from

If one radian is the angle standing on an arc of length $R$, then the number of radians in a full turn is just the number of times $R$ fits into the circumference:

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

So a full turn is $2\pi$ radians. Halve it:

$$360^\circ = 2\pi \text{ rad} \qquad\Longrightarrow\qquad \boxed{180^\circ = \pi}$$

Everything else follows from that one line. Divide both sides by $180$:

$$1^\circ = \frac{\pi}{180}\text{ rad}, \qquad 1\text{ rad} = \frac{180^\circ}{\pi} \approx 57^\circ 17' 45''$$

Notice that a radian is a big angle — nearly $60^\circ$. If a conversion ever comes out to ”$2$ radians is about $6^\circ$”, you have divided where you should have multiplied.

Converting, in practice

Degrees to radians: multiply by $\frac{\pi}{180}$.

$$75^\circ = 75\cdot\frac{\pi}{180} = \frac{5\pi}{12}$$

If the angle carries minutes, remember $30'$ is $\frac{30}{60}$ of a degree, so it contributes $30\cdot\frac{\pi}{180 \cdot 60}$:

$$82^\circ 30' = 82\cdot\frac{\pi}{180} + 30\cdot\frac{\pi}{180\cdot 60} = \frac{11\pi}{24}$$

Radians to degrees: don’t memorise a second rule. Just replace $\pi$ by $180^\circ$ — that is all “converting to degrees” means:

$$\frac{3\pi}{4} = \frac{3\cdot 180^\circ}{4} = 135^\circ, \qquad \frac{11\pi}{6} = \frac{11\cdot 180^\circ}{6} = 330^\circ$$

That substitution is worth internalising, because it turns every radians-to- degrees question into arithmetic you can do in your head. $\frac{\pi}{6}$? Replace $\pi$: $\frac{180^\circ}{6} = 30^\circ$. $\frac{5\pi}{3}$? $\frac{5 \cdot 180^\circ}{3} = 300^\circ$.

Why bother, if degrees work fine

Because past this point, the angle stops being a measurement and starts being a number you do calculus with. In radians, $\sin x \approx x$ for small $x$, the derivative of $\sin x$ is $\cos x$, and arc length is simply $s = R\theta$. All three pick up ugly constant factors in degrees. The circle does not care which unit you chose, but the formulas do.