The formula
radians = degrees × π/180
anchor: π radians = 180°, and one radian = 57.29578°
Type radians as multiples of pi (pi/6, 2pi/3) and the conversion is exact, because the π cancels: aπ/b radians is exactly a × 180/b degrees. Type a decimal and you get a decimal. Going the other way, this page hands common angles back as exact π fractions, since 45° is not approximately π/4, it is π/4.
Worked example
2π/3 radians = (2/3) × 180 = 120°, exactly. And 135° = 135/180 reduced = 3/4, so 3π/4 radians, exactly.
A decimal works too: 1 radian = 180/π = 57.29578°.
The table every trig class writes on the board
| Degrees | Radians | Worth remembering because |
|---|---|---|
| 30° | π/6 | sin 30° = 1/2, the cleanest fact in trigonometry |
| 45° | π/4 | the diagonal of a square |
| 60° | π/3 | every corner of an equilateral triangle |
| 90° | π/2 | the right angle |
| 180° | π | the anchor: a half turn |
| 270° | 3π/2 | three quarters of the way around |
| 360° | 2π | a full turn, and why 2π keeps appearing everywhere |
Why radians exist at all
Degrees are a human convention: Babylonian astronomers split the circle into 360 because their year was schematically 360 days, and the number survived because it divides beautifully. Radians are not a convention, they are the circle measuring itself: an angle of 1 radian is the angle you get when the arc along the rim is exactly as long as the radius. That definition makes arc length simply radius × angle, with no conversion constant, and it is why a full turn is 2π: the circumference is 2π radii, so the circle contains 2π of its own radius laid along its edge.
The payoff arrives in calculus, and it is the real reason the radian is not optional. The derivative of sin(x) is cos(x) only when x is in radians. Do calculus in degrees and every derivative drags a factor of π/180 behind it forever, like a tin can tied to the mathematics. Radians are the unit in which the circle's arithmetic comes out clean, which is why every formula past trigonometry quietly assumes them, and why the first bug in a thousand programs is a sine function fed degrees.
The mystery third button
Scientific calculators have a mode button reading D, R and G, and almost nobody alive has pressed G on purpose. It stands for gradians: the French Revolution's decimal angle, 400 to a full circle so that a right angle is a tidy 100. It failed for the same reason decimal time failed, but it never quite died; gradians survive in some European surveying, and so every calculator made since carries a button-sized monument to the revolution that could not beat Babylon. If your trig answers ever come out inexplicably wrong, check that mode button first: it is the most common calculator fault in every math class on Earth.
The mistake this page catches
Typing a decimal that means a pi multiple. 3.14159 radians converts to 179.99985°, not 180°, because 3.14159 is not π, it is a five-decimal souvenir of π. The gap seems small until it compounds through a calculation. When an exact angle is what you mean, type it as pi and let the symbol do its job; the whole point of π is that it carries infinite decimals without you having to.