Radians to degrees

To convert radians to degrees, multiply by 180/π. To convert degrees to radians, multiply by π/180. Both rules come from one fact: a full turn is 360° and also 2π radians, so 180° and π radians are the same angle^[1][2]. One radian is about 57.2958°, and one degree is about 0.017453 radians.

The two formulas, and why they work§

A degree is a convention: 360 of them to a full turn. A radian is the circle’s own unit. Bend the radius of any circle around its rim, and the angle that bent radius spans at the center is one radian. The rim is 2π radii long, so a full turn is 2π radians, about 6.283. The radians chapter shows this with a figure you can wrap.

Since 2π radians = 360°, dividing both sides by 2π gives 1 radian = 360°/2π = 180°/π ≈ 57.2958°, and dividing by 360 gives 1° = 2π/360 = π/180 ≈ 0.0174533 radians. established The International System of Units says the same thing in its table of units accepted for use with the SI: the degree is defined as (π/180) rad, the minute of arc as 1/60 of a degree and the second of arc as 1/60 of a minute^[2].

A good check on any conversion: the factor 180/π is about 57, so radians to degrees should make the number bigger by a factor of about 57, and degrees to radians should make it smaller. If your answer went the other way, you used the wrong fraction.

Worked examples§

150° to radians. Multiply by π/180: 150 × π/180 = 150π/180. Simplify the fraction by dividing top and bottom by 30: 5π/6. As a decimal, 5π/6 ≈ 2.618 radians. Leaving the answer as a multiple of π keeps it exact, and on the unit circle it tells you at a glance that the angle is π/6 short of a half turn.

3π/4 to degrees. Multiply by 180/π: (3π/4) × (180/π). The π cancels, leaving 3 × 180/4 = 135°. When the radian measure contains π, the π always cancels, which is why textbook answers come out as whole numbers.

2 radians to degrees. No π to cancel this time: 2 × 180/π = 360/π ≈ 114.59°. A radian measure without π is not wrong or unusual; it is just not a neat fraction of a turn. Calculus and physics produce radian measures like 2 or 0.5 all the time.

22.5° to radians. 22.5 × π/180 = 22.5π/180. Double the top and bottom to clear the decimal: 45π/360 = π/8. Every angle given to a finite number of decimal places is a rational multiple of π, and the converter above finds the fraction for you.

Common angles in degrees and radians§

And going the other way, for radian measures without π: 0.5 rad ≈ 28.65°, 1 rad ≈ 57.30°, 2 rad ≈ 114.59°, 3 rad ≈ 171.89° (just short of a half turn, since π ≈ 3.14), and 6 rad ≈ 343.77° (just short of a full turn, since 2π ≈ 6.28). These are the whole-radian ticks on the converter’s circle. The full chart of special angles, with their sines and cosines, is on the printable unit circle chart.

Three common mistakes§

The calculator is in the wrong mode. Most scientific calculators can work in degrees or radians, and they show the mode only as a small DEG or RAD on the screen. Type sin 30 in radian mode and you get about −0.988, the sine of 30 radians, instead of 0.5. established If a trig answer looks absurd, check the mode before anything else. Spreadsheets and programming languages almost always expect radians, so =SIN(30) in a spreadsheet gives −0.988 too; convert first, with =SIN(RADIANS(30)) or 30*PI()/180.

Using the fraction upside down. Multiplying 150° by 180/π gives about 8,594, which is no angle anyone means. The quick check from above catches it: degrees to radians must make the number smaller.

Forgetting that π is a number. “π/6” is an ordinary number, about 0.5236. It is tempting to treat π as a unit, like a degree sign, and write π/6 = 30. The equation that is true is π/6 radians = 30°. The radian is the unit; π is part of the amount. The confusion is common even among future teachers: in a study of 42 of them, Hatice Akkoç found two separate images of π, one an angle and the other an irrational number, held without being connected^[3]. documented

Degrees, minutes and seconds§

Navigation, surveying and astronomy often split a degree into 60 minutes of arc (′) and a minute into 60 seconds (″), the same base-60 system as hours, minutes and seconds^[2]. To convert a decimal degree, take the whole number of degrees, multiply what is left by 60 for the minutes, and multiply what is left after that by 60 for the seconds. One radian, 57.29578°, is 57° 17′ 44.8″: the 0.29578° left after 57 is 17.747′, and the 0.747′ left after 17 is 44.8″. The converter shows every angle this way as well.

Why mathematicians insist on radians§

If degrees are fine for everyday angles, why does every calculus book switch to radians? Because in radians the formulas lose their stray constants. The arc of a circle is s = rθ and the area of a sector is ½r²θ only with θ in radians; in degrees each picks up a factor of π/180. The same factor would appear in the derivative of sine: in radians it is cos x, in degrees it is (π/180) cos x. The radians chapter explains where that factor comes from and why it was worth reorganizing trigonometry around the radian.