Degrees to Radians

The formula, and why 180 degrees equals pi radians

Degrees and radians measure the same thing: how far an angle opens. One full turn is 360 degrees and also 2 pi radians, so half a turn is 180 degrees and pi radians. Divide both sides of that identity by 180 and one degree equals pi over 180 radians, which works out to 0.0174533.

The conversion runs both ways:

• degrees to radians: radians = degrees x pi / 180
• radians to degrees: degrees = radians x 180 / pi

You can see the identity on any circle. The circumference is 2 pi radii long, so a point that walks the whole circumference sweeps 2 pi radians of angle, and half of it sweeps pi radians, which is the same opening as a straight line, 180 degrees. One degree is therefore one 180th of pi.

Pi is the ratio of a circle's circumference to its diameter, 3.14159265 to eight decimals. The radian is the SI unit for plane angle, defined in the SI brochure published by the International Bureau of Weights and Measures, and the US conversion guidance sits in NIST Special Publication 811. Because pi never comes out even, degree values are exact and radian values are rounded to as many decimals as you need.

Worked example: 90 degrees in radians

Take 90 degrees. Multiply by pi: 90 x 3.14159265 = 282.7433385. Divide by 180: 282.7433385 / 180 = 1.5708 radians, or 1.5707963 to seven decimals. Because 90 is half of 180, the answer is also exactly pi over 2, which is why the converter prints both the decimal and the fraction form.

Two more, the ones that come up most often:

• 30 degrees: 30 / 180 = 1/6, so 30 degrees is pi/6 = 0.5236 rad
• 45 degrees: 45 / 180 = 1/4, so 45 degrees is pi/4 = 0.7854 rad

The pattern behind the fraction form is worth seeing. The multiple of pi is just the angle divided by 180, reduced to lowest terms. 30/180 = 1/6, 45/180 = 1/4, 90/180 = 1/2 and 135/180 = 3/4. That is why a unit circle in a textbook is labeled in sixths and quarters of pi rather than in decimals, and why memorizing 30, 45, 60 and 90 degrees covers most of the cases that show up in class.

Common angles in degrees and radians

This chart covers the angles that appear in trigonometry homework, geometry problems and code comments. The milliradian column divides the radian value by 0.001, which is the unit rifle sights and optics use.

Check one row by hand: 60 degrees is 60 x 3.14159265 / 180 = 1.0472, and the milliradian column is that value times 1,000. The sine and cosine columns are here because they only make sense in the right mode; a calculator in degree mode and one in radian mode will disagree about sin(30) even though both are working correctly.

Going the other way: radians to degrees

The reverse conversion multiplies by 180 and divides by pi, which is the same as multiplying by 57.2958. Four examples cover the common cases:

• 1 radian = 180 / pi = 57.2958 degrees
• 0.5 radians = 0.5 x 57.2958 = 28.6479 degrees
• 2 radians = 2 x 57.2958 = 114.5916 degrees
• pi radians = 180 degrees, by definition

A quick sanity check: an angle in radians is always the smaller number, because a radian is the bigger unit. 1.5708 radians and 90 degrees describe the same opening, and 1.5708 is smaller than 90. If your converted result is larger than the input and you expected the opposite, the input was probably already in degrees and got converted a second time.

Spreadsheets, code and calculators: the conversion already exists

You rarely have to type the constant, because most tools ship a function for it. The table shows what each one returns for 90 degrees.

In Excel and Sheets the reverse is =DEGREES(1.5708), which returns 90. The Microsoft RADIANS function page documents the syntax and the DEGREES counterpart, and Python's math module ships radians() and degrees() with the same behavior.

Spreadsheets keep full double precision internally, so =RADIANS(1) and 1 x PI()/180 agree to 15 digits; the display rounds, the stored value does not. The same is true of Python and JavaScript, which is why it pays to convert once at the edge of your data and leave the full precision value alone after that.

The one place the conversion bites people is the angle mode on a pocket calculator. A calculator left in DEG mode gives sin(30) = 0.5, which is correct for 30 degrees. In RAD mode the same keys give -0.9880, which is correct for 30 radians, about 4.8 full turns around the circle. Both answers are right; the mode decides which question was asked.

Why radians exist: arc length and the unit circle

A radian is defined so that arc length equals radius times angle: s = r x theta, with theta in radians. On a unit circle the radius is 1, so the arc length and the angle are the same number, and that is the whole reason the unit exists. Degrees need an extra conversion factor at every step; radians do not.

Worked example: a 60 degree angle on a circle with a 2 meter radius. Sixty degrees is 1.0472 radians, so the arc is 2 x 1.0472 = 2.0944 meters. The same setup gives the area of the sector: A = 0.5 x r squared x theta, so with a radius of 2 the working is 0.5 x 4 x 1.0472 = 2.0944 square meters.

The same math turns up on real objects. A 26 inch bike wheel has a 13 inch radius. Rotate it through 90 degrees and the contact point moves 13 x 1.5708 = 20.4204 inches along the ground; a full turn moves it 13 x 6.2832 = 81.6816 inches, which is the circumference of the wheel.

That clean relationship is why calculus, physics and graphics engines work in radians. The derivative of sine is cosine only when the angle is in radians; in degrees, an extra pi/180 factor shows up in every derivative. Angular velocity works the same way: one revolution per second is 2 pi radians per second, about 6.2832 rad/s, and one revolution per minute is 6.283185 / 60 = 0.1047 rad/s.

Everyday places the conversion shows up

Radians are not just a math class unit. They appear wherever software rotates or measures something round, and the same turn is often written one way in the interface and the other way in the code.

• CSS and design tools take degrees: transform: rotate(45deg) tilts an element by 45 degrees, which is 0.7854 radians.
• Canvas and WebGL take radians: ctx.rotate(Math.PI / 4) draws the same 45 degree turn, because the function reads radians.
• Game engines mix both: an inspector may show euler angles in degrees while the physics API wants radians, so every hand-off is a conversion.
• Photography uses radians for field of view: a 50 mm lens on a full-frame sensor covers 0.6911 radians horizontally, or 39.6 degrees.
• Navigation and astronomy compute in radians and print degrees, because a degree divided into 60 arcminutes is easier to read on a chart.

Spec sheets mix the units too. A camera gimbal quoted as having 0.35 radians of travel covers 0.35 x 57.2958 = 20.0535 degrees, and a drone that banks at 30 degrees is banking at 0.5236 radians.

Getting the unit wrong in a canvas call turns a small tilt into a 45 radian spin, about 7.16 full turns, which usually looks like the element vanishing. If a rotation looks wildly wrong in code but right in a preview panel, check which unit each one expects.

Rounding, negative angles and common mistakes

Radians are usually rounded to four decimals for display and kept at full precision for calculation. Rounding early is the usual source of small errors: 90 degrees rounded to 1.57 radians and converted back gives 1.57 x 180 / 3.14159265 = 89.954 degrees, a drift of 0.046 degrees that came entirely from rounding.

Negative angles follow the same rule with a minus sign: -90 degrees is -1.5708 radians, a quarter turn clockwise. Angles past a full turn are fine too: 450 degrees is 7.8540 radians, one and a quarter turns, and 720 degrees is 12.5664 radians, two full turns. The three mistakes to watch for are rounding before multiplying, entering a radian value into a degrees field, and leaving a calculator in the wrong angle mode.