How to Convert Degrees to Radians (and Back)

The conversion factor, and where it comes from

To learn how to convert degrees to radians, start with one fact: a half turn is 180° and it is also π radians. Those are two measurements of the same angle, so the ratio between the units is π/180.

That gives the whole method. Multiply a degree measure by π/180 to get radians; multiply a radian measure by 180/π to go back. There is nothing else to it — the only difficulty is remembering which way round the factor goes, and the half-turn fact settles that every time.

Converting degrees to radians

Multiply by π/180, then reduce the fraction. Leave π in the answer.

The reduction step is what turns an ugly fraction into a recognisable standard angle. 135π/180 and 3π/4 are the same number, but only the second is usable at a glance.

Converting radians to degrees

Multiply by 180/π. When the angle already contains π, the symbols cancel and the arithmetic is easy.

The standard angles worth memorising

A small table covers most of what appears in trigonometry, and recognising these on sight is faster than converting every time.

The unit circle ties the two together

The unit circle — a circle of radius 1 centred at the origin — is where radian measure stops being an arbitrary convention. Going a distance of 1 around its edge sweeps out exactly 1 radian. The angle and the arc length are the same number.

Because the full circumference is 2π, a full turn is 2π radians, half a turn is π radians, and a quarter turn is π/2. Every standard angle is a fraction of that circumference, which is why radian measures come out as fractions of π rather than round decimals.

Reading angle measure off the unit circle also explains why π/4 sits diagonally between the axes and why π/6 and π/3 are the two angles in a 30-60-90 triangle. The fractions are not arbitrary labels — they are how far around the circle you have travelled.

Arc length, the reason the definition pays off

For a circle of radius r, the arc cut by an angle θ has length rθ — provided θ is in radians. That is the whole formula.

In degrees the same calculation needs a conversion factor bolted on: arc length becomes 2πr × (θ/360). Both give the same answer, but only one of them is worth remembering, and the difference comes entirely from how the unit was defined.

The same simplification shows up in the area of a sector, in angular velocity, and everywhere else an angle multiplies a length.

Why radians exist at all

Degrees are arbitrary. Splitting a circle into 360 parts is a historical choice, convenient because 360 has many divisors, but nothing in the mathematics requires it.

A radian is defined by the circle itself: it is the angle that cuts an arc equal in length to the radius. That definition makes arc length simply rθ, with no conversion constant, and it is what makes calculus behave. The derivative of sin x is cos x only when x is measured in radians; in degrees, an awkward factor of π/180 appears and never goes away.

Keep the answer exact

3π/4 is exact. 2.356 is rounded, and the rounding compounds if the angle feeds into another calculation. Unless a decimal is specifically requested, leave π in the answer.

One practical warning that has nothing to do with the arithmetic: check which mode your calculator is in. sin(30) is 0.5 in degree mode and about −0.988 in radian mode. Both are correct answers to different questions, which is exactly what makes the error so easy to miss.

Questions about how to convert degrees to radians