To convert radians to degrees, multiply the radian measure by 180/π. With π in the angle, the π cancels and you get an exact answer: 5π/6 × 180/π = 150°. Without π, multiply by about 57.2958: 2 rad ≈ 114.59°. To go back from degrees to radians, multiply by π/180.
This guide shows why that factor works, a fast "slice" shortcut for angles like 7π/4, how to handle plain decimals such as 2.5 rad, a full table with reference angles for all four quadrants, and what to do with negative angles or angles bigger than one full turn.
π rad = 180°, so degrees = radians × 180/π and radians = degrees × π/180. One radian is 180/π ≈ 57.2958°.What Is a Radian?
Degrees split a full turn into 360 equal parts, a number chosen by convention. A radian is built from the circle itself: it's the angle at the centre that cuts off an arc exactly as long as the radius. OpenStax Precalculus (section 5.1) gives this definition and the result that follows from it: a full turn is 2π radians, because the circumference is 2π radii long.
So 360° = 2π rad, and halving both sides gives the one fact you need to remember: 180° = π rad. Everything else in this guide comes from that line.
Because a radian is arc length divided by radius, it's a ratio of two lengths. NIST's chart of SI units lists the radian as the SI derived unit of plane angle, equal to m/m = 1. That's why an angle written with no unit, like sin 2, means 2 radians, which OpenStax also states.
How to Convert Radians to Degrees
The factor 180°/π rad is a fraction whose top and bottom are the same angle, so it equals 1. Multiplying by it changes the units without changing the angle, the same way multiplying by 100 cm / 1 m turns metres into centimetres.
- Write the angle in radians
For example,
5π/6 rad. - Multiply by 180°/π
Put radians on the bottom of the factor so the radian unit (and the π) cancels:
(5π/6) × (180°/π). - Cancel and simplify
The π cancels, leaving
5 × 180°/6 = 900°/6 = 150°.
How do you remember which way up the factor goes? The unit you want goes on top. You want degrees, so 180° sits in the numerator. Going the other way, you want radians on top, so the factor is π/180°.
Convert 7π/4 radians to degrees.
Multiply by the factor: (7π/4) × (180°/π).
Cancel π: 7 × 180°/4.
Divide first to keep the numbers small: 180/4 = 45, so 7 × 45° = 315°.
7π/4 rad = 315°The Slice Shortcut for Fractions of π
Since π = 180°, you can replace π with 180° and simplify. Doing that to π/n gives the size of one "slice", and the numerator counts how many slices you have.
| Denominator | One slice | Example |
|---|---|---|
| 2 | π/2 = 90° | 3π/2 = 3 × 90° = 270° |
| 3 | π/3 = 60° | 4π/3 = 4 × 60° = 240° |
| 4 | π/4 = 45° | 5π/4 = 5 × 45° = 225° |
| 6 | π/6 = 30° | 11π/6 = 11 × 30° = 330° |
| 12 | π/12 = 15° | 7π/12 = 7 × 15° = 105° |
This is the same calculation as multiplying by 180/π, just in a friendlier order. It also tells you where the angle lands: with denominator 6, twelve slices make a full turn, so 11π/6 is one slice short of 360°.
Converting Radians Without π
Angles such as 2.5 rad or 0.8 rad come up in calculus, physics and calculator output. There's no π to cancel, so the answer is a decimal. Multiply by 180/π and round at the very end.
Convert 2.5 rad to degrees, and say which quadrant the angle is in.
Multiply: 2.5 × 180/π = 450/π. This is the exact value.
Evaluate: 450 ÷ 3.14159… ≈ 143.24°.
Quadrant: π/2 ≈ 1.571 and π ≈ 3.142, and 2.5 sits between them, so the angle is in Quadrant II. The degree answer agrees: 143.24° is between 90° and 180°.
2.5 rad = 450/π ≈ 143.24°, Quadrant IIShould you leave the answer as 450/π or write 143.24°? If the radian measure had π in it, give the exact degree value. If it was a plain decimal, a rounded decimal is expected; two decimal places is a common choice unless your teacher asks for something else.
Replacing π with 180 only works when π is there to replace. Students sometimes "convert" 2.5 rad by writing 2.5 × 180 = 450°, which is more than a full turn. The 180 has to be divided by π: 2.5 rad is about 143°, not 450°.
Degrees to Radians: Going the Other Way
Multiply by π/180, then reduce the fraction. Dividing the degree value and 180 by their greatest common factor is the cleanest way to do it.
Convert 225° and 40° to radians.
225° × π/180° = 225π/180. The greatest common factor of 225 and 180 is 45, so this reduces to 5π/4.
40° × π/180° = 40π/180. The greatest common factor is 20, giving 2π/9.
Decimal check: 2π/9 ≈ 0.698, a bit less than π/4 ≈ 0.785, which fits because 40° is a bit less than 45°.
225° = 5π/4 and 40° = 2π/9If reducing fractions is rusty, the fraction calculator will simplify 225/180 and show the steps.
Degrees, Radians and Reference Angles Table
The unit circle uses sixteen standard angles from 0° to 330° (360° repeats 0°). Each one has a reference angle: the acute angle between its terminal side and the x-axis. That reference angle is always 30°, 45° or 60° (or π/6, π/4, π/3) for the non-quadrantal angles, which is why you only memorise the first quadrant.
| Degrees | Radians | Quadrant | Reference angle |
|---|---|---|---|
| 0° | 0 | on the positive x-axis | none |
| 30° | π/6 | I | 30° (π/6) |
| 45° | π/4 | I | 45° (π/4) |
| 60° | π/3 | I | 60° (π/3) |
| 90° | π/2 | on the positive y-axis | none |
| 120° | 2π/3 | II | 60° (π/3) |
| 135° | 3π/4 | II | 45° (π/4) |
| 150° | 5π/6 | II | 30° (π/6) |
| 180° | π | on the negative x-axis | none |
| 210° | 7π/6 | III | 30° (π/6) |
| 225° | 5π/4 | III | 45° (π/4) |
| 240° | 4π/3 | III | 60° (π/3) |
| 270° | 3π/2 | on the negative y-axis | none |
| 300° | 5π/3 | IV | 60° (π/3) |
| 315° | 7π/4 | IV | 45° (π/4) |
| 330° | 11π/6 | IV | 30° (π/6) |
To find a reference angle in radians, use the distance to the nearest part of the x-axis: π − θ in Quadrant II, θ − π in Quadrant III and 2π − θ in Quadrant IV. For 5π/6, that's π − 5π/6 = π/6, matching the 30° in the table.
Negative Angles and Angles Over 360°
The conversion factor doesn't care about size or sign: −π/3 is −60°, and 4π is 720°. What changes is the follow-up. To place a big or negative angle on the unit circle, add or subtract 360° (or 2π) until the result lies between 0° and 360°. That result is called a coterminal angle, because it ends on the same terminal side.
Convert 23π/6 to degrees, then find its coterminal angle in [0°, 360°) and its reference angle.
Slice method: the denominator is 6, so each slice is 30°. 23 × 30° = 690°.
Subtract one full turn: 690° − 360° = 330°. In radians, 23π/6 − 2π = 23π/6 − 12π/6 = 11π/6.
330° is in Quadrant IV, so the reference angle is 360° − 330° = 30°, or π/6.
23π/6 = 690°, coterminal with 330° (11π/6), reference angle 30°Why Radians Make Arc Length Simple
If radians are this much trouble, why not use degrees everywhere? Because the definition of a radian makes arc length a single multiplication: s = rθ, with θ in radians. OpenStax gives the same formula in section 5.1. In degrees, the same arc needs a fraction of the circumference: s = (θ/360°) × 2πr. Calculus leans on radians for a related reason: the derivative of sin x is cos x only when x is in radians.
A wiper blade is 45 cm long and sweeps through 120°. How far does its tip travel in one sweep?
Convert the angle: 120° × π/180° = 2π/3 rad.
Arc length: s = rθ = 45 cm × 2π/3 = 30π cm. The radian unit is just the number 1, so the answer is in centimetres.
Evaluate: 30π ≈ 94.2 cm.
Same answer in degrees: (120/360) × 2π × 45 = (1/3) × 90π = 30π cm.
30π cm ≈ 94.2 cmUsing s = rθ with θ still in degrees. Here that gives 45 × 120 = 5400 cm, about 54 m for one wipe. The formula only holds in radians, so convert first. The same slip happens on a calculator left in the wrong mode: sin 30 is 0.5 in degree mode, but in radian mode it's sin(30 rad) ≈ −0.988. To test the mode, type sin 30: 0.5 means degree mode.
How to Check Your Answer
Three quick checks catch most conversion slips:
- Size estimate. One radian is a bit under 60°, and π is a bit over 3. So 2.5 rad should be a bit under
2.5 × 60° = 150°. If you got 0.04°, the factor is upside down; if you got 450°, you multiplied by 180 without dividing by π. - Landmarks. Compare with
π/2 ≈ 1.57,π ≈ 3.14,3π/2 ≈ 4.71and2π ≈ 6.28. A radian value between 3.14 and 4.71 must convert to something between 180° and 270°. - Convert back. Multiply your degree answer by
π/180. You should land on the radian measure you started with:315° × π/180 = 7π/4.
Once you've worked a problem by hand, Solver AI can check it: snap a photo of the question and compare its step-by-step solution with your own lines to see where they split.
Everything comes from π rad = 180°. Put the unit you want on top of the factor, swap π for 180° when it's there, divide by π when it isn't, and add or subtract a full turn to place the angle on the unit circle.
Practice Problems
- Convert 7π/12 to degrees
Answer: one slice is
π/12 = 15°, so7 × 15° = 105°. - Convert 3.6 rad to degrees and give the quadrant and reference angle
Answer:
3.6 × 180/π = 648/π ≈ 206.26°. That's between 180° and 270°, so Quadrant III, with reference angle206.26° − 180° ≈ 26.26°. - Convert 330° to radians
Answer:
330π/180; the greatest common factor is 30, so11π/6. - Convert −11π/4 to degrees and find the coterminal angle in [0°, 360°)
Answer:
−11 × 45° = −495°. Add two full turns:−495° + 720° = 225°(5π/4), in Quadrant III with reference angle 45°. - A fan blade tip 0.5 m from the centre travels 20 m of arc. Through what angle did it turn?
Give the angle in radians and degrees, the number of complete turns, and where the tip ends up. Answer:
θ = s/r = 20 m ÷ 0.5 m = 40 rad, which is40 × 180/π = 7200/π ≈ 2291.83°. Six full turns are 2160°, and2291.83° − 2160° ≈ 131.83°, so the tip makes 6 complete turns and ends at about 131.83°, in Quadrant II with reference angle about 48.17°.
Next, use these angles to find exact sine and cosine values with the unit circle, or see radians at work in graphing sine and cosine functions, where the period 2π sets the length of one cycle. For more trig help, the trigonometry solver page covers the wider topic.