Radians

Where did 360 even come from?

Why does a full turn have 360 degrees, and not 100, or 1000? Nobody worked it out from the circle — it was chosen, thousands of years ago. The best guess: the Babylonians counted in base 60 (which is where our 60 minutes and 60 seconds come from too), and their year was roughly 360 days, so the Sun crept about one "degree" around the sky each day. Handy, human, and completely arbitrary.

A mathematician wants an angle unit the circle picks for itself. That unit is the radian. Draw a circle, take its radius, and bend that exact length around the rim like a piece of string. The angle you've just swept out — from the centre — is one radian (about 57.3^\circ). No committee, no calendar: the circle defines it using nothing but its own radius.

A circle's own unit of angle

Since the whole circumference is 2\pi r — that's 2\pi radius-lengths of string laid end to end around the rim — a full turn must be 2\pi radians. That gives the one fact everything else hangs on:

360^\circ = 2\pi \text{ radians}, \qquad \text{so } 180^\circ = \pi \text{ radians}.

That single line is the whole conversion. To go degrees → radians, multiply by \dfrac{\pi}{180}; to go radians → degrees, multiply by \dfrac{180}{\pi}. (Both fractions are just \tfrac{\pi\text{ rad}}{180^\circ} turned the way you need — the unit you want ends up on top.)

The values worth knowing by heart

From 180^\circ = \pi the common angles fall out by simple division — you rarely need the calculator for these:

a full turn 2\pi \text{ rad} = 360^\circ, and a half turn \pi \text{ rad} = 180^\circ;
common values: 30^\circ = \tfrac{\pi}{6}, 45^\circ = \tfrac{\pi}{4}, 60^\circ = \tfrac{\pi}{3}, 90^\circ = \tfrac{\pi}{2};
arc length s = r\theta and sector area A = \tfrac{1}{2}r^2\theta — but only when \theta is measured in radians.

Look how tidy those last two formulas are. Arc length is just "radius times angle"; sector area is just "half radius-squared times angle". In degrees you'd have to bolt an ugly \tfrac{\pi}{180} onto both. Radians make the circle's own measurements come out clean — that's the point of them.

See one radian

Step through the figure: lay down the radius, wrap an arc of the same length to sweep one radian, then continue to the half-circle, which is \pi radians. Watch the arc and the radius: they're always equal for a one-radian sweep — that's the definition, drawn.

Worked example 1 — converting the common angles

Turn 90^\circ, 60^\circ and 45^\circ into radians. Multiply each by \tfrac{\pi}{180} and cancel:

90^\circ = 90 \times \tfrac{\pi}{180} = \tfrac{90\pi}{180} = \tfrac{\pi}{2}.

60^\circ = \tfrac{60\pi}{180} = \tfrac{\pi}{3}, \qquad 45^\circ = \tfrac{45\pi}{180} = \tfrac{\pi}{4}.

The trick is to leave the answer as a fraction of \pi, not a decimal. \tfrac{\pi}{2} is exact and instantly recognisable; 1.5708\ldots is neither. Going the other way, to convert \tfrac{5\pi}{6} back to degrees, multiply by \tfrac{180}{\pi}: \tfrac{5\pi}{6}\times\tfrac{180}{\pi} = \tfrac{5\times180}{6} = 150^\circ.

Worked example 2 — arc length and sector area

A slice of pizza has a radius of 15\text{ cm} and its two straight edges meet at an angle of 0.8 radians. How long is the curved crust, and how much pizza is in the slice?

With \theta already in radians the formulas are direct:

s = r\theta = 15 \times 0.8 = 12\text{ cm of crust}.

A = \tfrac12 r^2\theta = \tfrac12 \times 15^2 \times 0.8 = \tfrac12 \times 225 \times 0.8 = 90\text{ cm}^2.

Notice how painless that was — because the angle was in radians. If the slice had been given as 45.8^\circ, step one would be to convert it to radians, because s = r\theta simply does not work in degrees.

Worked example 3 — a decimal conversion

Not every angle is a neat fraction of \pi. Convert 50^\circ to radians as a decimal:

50^\circ = 50 \times \tfrac{\pi}{180} = \tfrac{50\pi}{180} = \tfrac{5\pi}{18} \approx 0.87\text{ rad}.

Keep the exact \tfrac{5\pi}{18} if you can, and only fall back on the rounded 0.87 when a numerical answer is asked for.

Radians trip people up in three reliable ways:

Your calculator's MODE. This is the single biggest cause of wrong trig answers. Ask a calculator for \sin 30 while it is in radian mode and it thinks you mean 30 radians (nearly five full turns!) and returns -0.988, not the 0.5 you wanted. Before any trig calculation, check the little DEG / RAD flag on the screen and make sure it matches the units in the question.
Keep \pi in exact answers. Write \tfrac{\pi}{6}, not 0.524, unless a decimal is specifically requested. The exact form is shorter, precise, and shows you know the value.
s = r\theta and A = \tfrac12 r^2\theta are radians-only. Plug a degree value straight into them and the answer is nonsense. Convert to radians first, every time.

Here is the deep reason radians aren't just tidier — they're the only unit maths really wants. It's a beautiful fact that the derivative of \sin x is exactly \cos x:

\frac{d}{dx}\sin x = \cos x.

But that clean result is only true when x is in radians. Measure angles in degrees and an ugly stowaway appears: \frac{d}{dx}\sin x^\circ = \tfrac{\pi}{180}\cos x^\circ. Every derivative, every wave, every oscillation would drag that \tfrac{\pi}{180} around forever. So all of higher maths, physics and engineering quietly agrees to use radians and never look back — you'll rely on this the moment you meet the derivatives of sine and cosine.

One last provocation: since a full turn is 2\pi radians, some mathematicians argue that 2\pi (which they nickname \tau, "tau") is the circle's true constant, and that plain \pi — half a turn — is the awkward one. A quarter turn would just be \tfrac{\tau}{4}. The debate is half serious, half fun — and there's even a "Tau Day" on the 28th of June.