Radians to Degrees

The formula: multiply by 180 and divide by pi

Converting radians to degrees takes one multiplication and one division: degrees = radians × 180 ÷ π. Because pi never ends, most people use the decimal form instead:

• radians to degrees: degrees = radians × 57.2958
• degrees to radians: radians = degrees × 0.0174533

The 180 in the formula is not an invented constant. A half turn is 180 degrees and it is also exactly π radians, so those two numbers describe the same opening. Divide 180 by pi and you get the degree value of a single radian: 180 ÷ 3.14159265 = 57.2958, and the exact value, written out, is 57.29577951308232.

The radian is the SI unit for plane angle, listed in the SI Brochure, 9th edition, 2019 published by the International Bureau of Weights and Measures, which also states that a right angle is π/2 rad and that the radian can be written as m/m. The US National Institute of Standards and Technology keeps the same conversion guidance in NIST Special Publication 811 (2008) and on its SI units reference pages. Because the factor comes straight from the definition, it is exact, not something someone measured.

Worked examples, checked by hand

Take 2 radians. Multiply by 180: 2 × 180 = 360. Divide by pi: 360 ÷ 3.14159265 = 114.5916 degrees. That is the whole method. The same three steps give every value in this list:

• 1 radian: 180 ÷ 3.14159265 = 57.2958 degrees
• 0.5 radians: 0.5 × 57.2958 = 28.6479 degrees
• 1.5 radians: 1.5 × 57.2958 = 85.9437 degrees
• π/4, which is 0.78539816 radians: 0.78539816 × 57.2958 = 45.0000 degrees, a quarter turn
• 10 radians: 10 × 57.2958 = 572.9578 degrees, about 1.59 turns around a circle

A fast sanity check runs on the size of the numbers, not the arithmetic: a degree is the smaller unit, so the degree value of an angle is always the larger number. If you convert 2 radians and the answer comes out smaller than 2, something went wrong, because 2 radians is nearly a third of a full turn.

Conversion table: common radian values

The rows below cover the angles that show up in homework, datasheets and code. The turns column is the radian value divided by 2π, so 1 tells you a full rotation.

Check one row the long way: 3 radians is 3 × 180 ÷ 3.14159265 = 171.8873 degrees. The π rows land on round numbers because those radian values are a quarter, a half and a full turn by definition.

The degrees minutes seconds column is worked out from the exact angle, not from the rounded decimal. Type 1.5708 into the calculator and it prints 90° 0′ 0.8″ instead of 90° 0′ 0.0″, because 1.5708 is a touch more than π/2; the 0.0002 degree gap is the rounding, not a mistake in the conversion.

Where the number 57.2958 comes from

Pi is the ratio of a circle's circumference to its diameter, 3.14159265 to eight decimals. A full turn sweeps the whole circumference, so a full turn is 2π radians and also 360 degrees. Half of that identity gives 180 degrees = π radians, and dividing both sides by π gives 1 radian = 180/π = 57.29577951 degrees. Since pi is irrational, the decimal runs forever, which is why every printed value stops somewhere and why the calculator keeps more digits than it shows.

In plain words, a radian is the angle you get when you bend the radius of a circle along the rim: the arc spans one radius, the angle at the center is one radian. That is the definition the SI Brochure uses, and it is also why arc length in radians needs no extra factor: s = r × θ holds only when θ is in radians.

The degree, meanwhile, is a historical unit from Babylonian astronomy, defined today as π/180 rad. So the pair of factors 57.2958 and 0.0174533 are two ways of writing the same exact relationship, 180/π and its reciprocal.

Degrees, minutes and seconds

Navigation, surveying and astronomy split a degree into 60 arcminutes and each arcminute into 60 arcseconds, written 114° 35′ 29.6″. The conversion is the same, just with a second formatting step. Take 2 radians, which is 114.59155903 degrees:

• Whole degrees: 114
• Leftover 0.59155903 × 60 = 35.4935418, so 35 arcminutes
• Leftover 0.4935418 × 60 = 29.61, so about 29.6 arcseconds

So 2 radians is 114° 35′ 29.6″. The same steps turn 1 radian into 57° 17′ 44.8″ and 0.5 radians into 28° 38′ 52.4″. The calculator above prints this form next to the decimal answer so you do not have to repeat the three multiplications.

GPS devices often use a halfway form, decimal minutes: 114.59155903 degrees is 114° 35.4935′. Boaters and pilots see that variant on charts, and it converts with one multiplication instead of two.

Where the conversion shows up

Software and hardware mix the two units constantly, and the same rotation is often typed one way in a settings panel and stored the other way in code:

• Pocket calculators carry a DEG/RAD mode. In degree mode sin(30) = 0.5; in radian mode the same keys give −0.9880, which is correct for 30 radians, almost 4.8 full turns. The mode decides which question got asked.
• CSS takes degrees: rotate(45deg) tilts an element 45 degrees. Canvas and WebGL take radians: ctx.rotate(0.7854) draws the same tilt.
• Spec sheets quote motion in radians: a camera gimbal with 0.35 rad of travel covers 0.35 × 57.2958 = 20.0535 degrees.
• Angular speed works the same way: a motor spinning at 10 rad/s sweeps 10 × 57.2958 = 572.9578 degrees per second, about 1.59 revolutions.

When a rotation looks wildly wrong in code but fine in a preview panel, the usual suspect is a radian value fed to a function that wanted degrees, or the reverse. Converting with 57.2958 by hand takes seconds and settles it.

Spreadsheets and code do it for you

Every common tool ships the conversion, so you rarely need to type 57.2958. The table shows what each one returns for 1 radian.

The reverse functions exist too: Excel's =RADIANS(90) returns 1.570796327, and Python's math.radians(90) returns 1.5707963267948966. Spreadsheets keep full double precision internally and only round what they display, so the stored value of =DEGREES(1) and 180/PI() agree to 15 digits.

JavaScript is the odd one out: there is no Math.degrees, so the expression 180 / Math.PI lives in a surprising amount of production code. Wrapping it in a small helper function once, a function that takes radians and returns them times 180 divided by pi, beats retyping the constant and mistyping it at 57.9258 or 57.2598.

Rounding, negative values and common mistakes

Four decimals are enough to read an answer, but keep full precision while calculating. Rounding π/2 to 1.5708 and converting back gives 1.5708 × 57.29577951308232 = 90.0002 degrees, a drift of 0.0002 degrees that came entirely from the rounding. Rounding 1 radian to 57.3 degrees throws away 0.0042 degrees, which matters on a surveying site even if it does not in a sketch.

Negative angles keep the sign: −2 radians is −114.5916 degrees, a clockwise turn. Angles past a full turn are fine too, so 7 radians is 401.0705 degrees, which is 41.0705 degrees past one full circle.

Three mistakes cover most of the errors people make: rounding before multiplying instead of after, typing a degree value into a field that expects radians, and leaving a calculator in the wrong angle mode. If a result looks off by roughly a factor of 57, a unit got swapped somewhere.