Radians to Degrees Calculator – calculatorpeak.com

About the Radians to Degrees Calculator

There are two ways to measure an angle, and only one of them is natural.

Degrees divide a circle into 360 parts. The number is inherited from Babylonian astronomy and their base-60 arithmetic, and it survives because 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more — which makes mental arithmetic easy. But nothing in mathematics requires it. It is a convention, and a very old one.

Radians divide a circle by the circle itself. One radian is the angle you get when the arc along the edge is exactly as long as the radius. Since the circumference is 2πr, exactly 2π radians fit around a circle — about 6.283.

That definition sounds abstract and turns out to be the reason radians run all of higher mathematics. This calculator converts between both, plus gradians, turns and arcminutes, and shows the exact multiple of π wherever there is one.

How to Use the Radians to Degrees Calculator

Pick the direction:

Radians to degrees, degrees to radians, multiples of π to degrees — for when a question gives you "3π/4" and you want a number you can picture — or gradians and turns for the less common units.

Every result shows the angle in all the units at once, so a single calculation answers whichever form you actually need.

The Conversion

Everything comes from one equality:

  π radians = 180°

Rearranged:

  degrees = radians × 180 / π
  radians = degrees × π / 180

  1 radian  = 57.29577951°
  1 degree  = 0.01745329 radians

The number worth memorising is 57.3° per radian. A radian is a bit under a sixth of a circle — large enough to picture, which is more than can be said for 0.01745.

The Angles You Will Meet

| Degrees | Radians | Exact | |---------|---------|-------| | 0° | 0 | 0 | | 30° | 0.5236 | π/6 | | 45° | 0.7854 | π/4 | | 60° | 1.0472 | π/3 | | 90° | 1.5708 | π/2 | | 120° | 2.0944 | 2π/3 | | 180° | 3.1416 | π | | 270° | 4.7124 | 3π/2 | | 360° | 6.2832 | 2π |

The right-hand column is why mathematicians write angles in radians: π/6 is exact and short, where 0.5236 is a rounded decimal that will never quite close a calculation. This calculator gives the exact form whenever the angle has one.

Why Radians Are the Natural Unit

Here is the fact that settles it. In radians:

  d/dx sin(x) = cos(x)

Clean. In degrees, the same derivative is:

  d/dx sin(x°) = (π/180) × cos(x°)

An awkward constant appears and then propagates through every subsequent line of working. The same thing happens to the small-angle approximation (sin x ≈ x, only true in radians), to Taylor series, and to every physics formula involving rotation.

Angular velocity in radians per second multiplies straight by the radius to give linear speed. In degrees per second it does not. Arc length is simply s = rθ in radians; in degrees you need a conversion factor.

None of this makes degrees wrong. For surveying, navigation, construction and everyday description, degrees are easier to say and easier to picture. Use degrees to communicate; use radians to calculate.

The Other Units

Gradians put 100 in a right angle and 400 in a circle, so a quarter turn is a round 100 — convenient for decimal surveying work. Many calculators still have a GRAD mode, and leaving it switched on by accident is a classic source of wrong trigonometry answers. If your sine of 45 comes out as 0.6494 rather than 0.7071, check the mode — 45 gradians is only 40.5°.

Turns are the simplest unit of all: one turn is one full circle. Used in rotational engineering, and quietly the most intuitive of the lot.

Arcminutes and arcseconds subdivide the degree by 60 and then 60 again, the same way as hours. Navigation and astronomy use them, and they are why a position reads as 51° 30′ 26″. This calculator shows that form alongside the decimal.

Step-by-Step Example

Radians to degrees. Convert 2.5 radians.

  2.5 × 180 / π = 2.5 × 57.29578 = 143.2394°

Just under 143°, so a bit past a right angle and a half. In quadrant 2.

Degrees to radians. Convert 135°.

  135 × π / 180 = 0.75π = 2.356194 radians

And note the exact form: 3π/4. That is the answer to write down if the calculation continues.

A multiple of π. What is 5π/6 in degrees?

  5/6 × 180 = 150°

More than one turn. Convert 800°.

  800 / 360 = 2 full turns, remainder 80°
  So 800° points the same way as 80°.
  In radians: 800 × π/180 = 13.9626

Angles beyond 360° are perfectly valid — they describe rotation rather than direction — but for anything about where something points, only the remainder matters.

Understanding Your Result

The result is the conversion you asked for.

The exact form gives the multiple of π when the angle has a clean one, and says plainly when it does not, rather than printing something like 0.6457π and implying more elegance than is there.

The all units line gives degrees, radians, gradians, turns and degrees-minutes-seconds together.

The where it points line strips out any full turns and names the quadrant, including which trigonometric ratios are positive there — useful when the conversion is a step inside a trigonometry problem.

When Should You Use This Calculator?

Trigonometry homework. Questions mix the units freely, and marks are lost on the conversion rather than the trigonometry.

Programming. JavaScript, Python, C and nearly every other language expect radians in sin, cos and tan. A common bug is passing degrees; the code runs and the answer is silently wrong.

Physics and engineering. Angular velocity, oscillation and wave equations all use radians.

Spreadsheets. Excel and Google Sheets have RADIANS() and DEGREES() for this reason — their trig functions take radians only.

CAD and 3D graphics. Rotations are usually stored in radians and displayed in degrees.

Navigation and astronomy. Positions come in degrees, minutes and seconds, but the mathematics behind them runs in radians.

Surveying. Where gradians may still appear, and need converting to something the rest of the world uses.

Common Mistakes

Leaving the calculator in the wrong mode. The single biggest cause of wrong trigonometric answers. sin(30°) is 0.5; sin(30 radians) is −0.988. Neither is an error message — both look like answers.

Multiplying when you should divide. Degrees to radians multiplies by π/180; radians to degrees multiplies by 180/π. Getting it backwards is out by a factor of about 3283.

Rounding π too early. Use the full value and round once at the end.

Assuming radians must contain π. 2 radians is a perfectly ordinary angle, about 114.6°. Only some angles have a neat π form.

Treating 2π and 360° as different angles. They are the same rotation written two ways.

Forgetting that minutes here are not decimals. 30′ is half a degree, not 0.30 of one. 51.5° is 51° 30′, not 51° 50′.

Passing degrees to a programming language. Math.sin(90) in JavaScript gives 0.894, not 1 — because 90 is being read as radians.