How do you convert degrees to radians?

The 180 is not arbitrary. A radian is the angle you get when the arc of a circle is exactly as long as its radius. The circumference is 2π times the radius, so 2π of those arcs fit around the circle, and a full turn is 2π radians. A full turn is also 360°, which is where every number on this page comes from:

360°   =  2π rad
180°   =  π rad
1°     =  π/180 rad    =  0.017453292519943295… rad
1 rad  =  180/π °      =  57.29577951308232… °

The degree is a human invention, 360 of them to a turn because 360 divides nicely. The radian is what the circle hands you if you measure angles with its own radius. That is why the trig functions in calculus, and inside every calculator, work in radians, and why the conversion has a π in it that never goes away.

The two directions

Degrees to radians: multiply by π/180. Sixty degrees:

60 × π/180   =  60π/180   =  π/3 rad   =  1.0471975511965977… rad   →  1.0472 rad

Radians to degrees: multiply by 180/π. Two radians:

2 × 180/π    =  360/π     =  114.59155902616464… °   →  114.59°

Both decimals run forever, because π does. The π/3 in the first line is the only exact thing on it, which is why the next section exists.

Keep it as a multiple of π

Write the degrees over 180, leave the π alone, and cancel the fraction. The π is not an inconvenience to be multiplied out. It is the answer:

120°   =  120π/180   =  2π/3       (divide top and bottom by 60)
135°   =  135π/180   =  3π/4       (by 45)
210°   =  210π/180   =  7π/6       (by 30)
270°   =  270π/180   =  3π/2       (by 90)
37°    =  37π/180    =  37π/180    (37 is prime, nothing cancels)

The fraction tells you what the angle is. 2π/3 is two thirds of a half turn, so 120° is two thirds of the way from 0 to 180, and you can see that on a clock face. 2.0944 tells you nothing you can check without a calculator. Cancelling is the same greatest common factor routine as simplifying any fraction: 120 and 180 share 60, so 120/180 is 2/3.

When the fraction will not cancel, as with 37°, the decimal is the more useful form. Multiply by 0.017453292519943295 and round:

37 × 0.017453292519943295   =  0.6457718232379019… rad   →  0.6458 rad

Radians to degrees

Multiply by 180/π. If the angle is written with a π, the π cancels and you are left with 180 divided by whatever was under it:

π/5    ×  180/π   =  180/5    =  36°
π/12   ×  180/π   =  180/12   =  15°
5π/4   ×  180/π   =  900/4    =  225°
7π/6   ×  180/π   =  1260/6   =  210°

A plain decimal number of radians has no π to cancel, so the 180/π stays as 57.29577951308232 and the answer is never round:

1 rad     =  57.29577951308232°   =  57° 17′ 44.8″
1.5 rad   =  85.94366926962348°    →  85.94°
3 rad     =  171.88733853924696°   →  171.89°

Three radians is a little under a half turn, which is a decent sanity check for any answer: π radians is 180°, so a radian figure just under 3.14 should give a degree figure just under 180.

In your head: divide by 57.3

Since a radian is 57.2958°, dividing degrees by 57.3 is 0.007% low, which is as good as any mental method deserves to be. Going back, multiply radians by 57.3, 0.007% high:

37°     →  37 ÷ 57.3    =  0.6457 rad     exact 0.6458
2 rad   →  2 × 57.3     =  114.6°         exact 114.59

If dividing by 57.3 in your head is not a thing you do, multiply degrees by 7 and divide by 400 instead, which is 0.0175 per degree and 0.27% high: 37° becomes 0.6475 rad against the true 0.6458. The trap to avoid is 60. A radian looks like it should be 60° and it is not. 60° is π/3, which is 1.047 rad, and dividing degrees by 60 runs 4.5% low. Whichever shortcut you take, round once, at the end, per how to round decimals.

The DEG/RAD trap

Every scientific calculator has a mode switch, usually labelled DEG and RAD, and most wrong trig answers come from it being set to the other one. The rule is to match the mode to the input. An angle like 30 or 45 is degrees, so the calculator must say DEG. An angle written with π, like π/6, is radians, so it must say RAD. In the wrong mode the calculator does exactly what it was told, with a number you did not mean:

sin(30)  in DEG   =  0.5
sin(30)  in RAD   =  −0.9880316240928618      (30 rad  =  1718.87°  =  4 turns + 278.87°)
cos(60)  in RAD   =  −0.9524129804151563
tan(45)  in RAD   =  1.6197751905438615

Thirty radians is nearly five full turns, and the sine of whatever is left over after four of them, 278.87°, is −0.988. The number is not wrong. The mode was. Prism keeps the DEG/RAD chip next to the display rather than in a menu, so the mode is visible on every result, and it starts in degrees. Some calculators carry a third mode, GRAD, for gradians: 400 to a turn, so 100 grad is a right angle and 1 grad is 0.9°, or π/200 rad. Unless you are a surveyor, that is the mode you set by accident.

Conversion table: degrees to radians

Every entry in the middle column is exact. The decimals are rounded to four places:

Conversion table: radians to degrees

Where binary arithmetic slips

A computer cannot hold π, so it holds the nearest double, which is about 1.2 × 10⁻¹⁶ short of the real thing. Then you ask it for the sine of 180°. It converts to radians, lands on its slightly short π, and takes the sine of that:

sin(180°)   in binary floating point   =  1.2246467991473532e-16
sin(180°)   exactly                    =  0

cos(90°)    in binary floating point   =  6.123233995736766e-17
tan(90°)    in binary floating point   =  1.633123935319537e16        (should be undefined)

The sine is not quite zero and the tangent, which should not exist at all, comes out as sixteen quadrillion. It is the same fault as 0.1 + 0.2 = 0.30000000000000004 in a different costume: the input was not the number you typed. Prism treats any trig result smaller than 10⁻¹² as the zero it is, snaps results within 10⁻¹³ of a whole number to that whole number, and keeps every function result to the 15 significant digits a double can actually vouch for. So sin(180°) is 0, tan(90°) is an error rather than 1.6e16, sin(30°) is 0.5 and not 0.49999999999999994, and 0.1 + 0.2 is 0.3.

Or flip the DEG/RAD chip and let Prism hold the π

Prism's scientific mode has the DEG/RAD toggle beside the display, a π key, and sin, cos, tan with their inverses. Scientific mode is free, like fractions, percentages and history. No ads, no subscription. See the app.