Angle Conversion Calculator

Created by Gabriela Diaz
Last updated: Jul 06, 2022

Use the angle conversion calculator to instantly convert radians to degrees, degrees to gradians, or many other angular units.

If you're curious about how to convert degrees to radians or degrees minutes seconds to decimal degrees, then we highly recommend reading the text below, where you'll also find:

  • The degrees to radians formula;
  • Its counterpart, the radians to degrees formula;
  • Why some angular measurements are expressed in degrees minutes seconds; and
  • An example of how to convert degrees minute seconds to decimal degrees.

In physics, a correct angle is important, e.g., in the diffraction phenomenon. If you're interested in that topic, our diffraction grating calculator is a great tool to start with!

How to convert degrees to radians?

The radian is the most often used unit for measuring angles, as it is the preferred angular measure in calculus, geometry, and many other fields of mathematics and engineering, as well as the SI unit for angles.

The radian unit is derived from the idea of measuring angles with the length of the arc. The magnitude of the subtended angle in radians is equal to the ratio of the arc length to the radius of the circle:

θ=s/r\small \theta = s/r

where:

  • θ — Angle in radians;
  • s — Arc length; and
  • r — Radius.

The radian symbol is the "rad", and 1 radian is equivalent to 57.2958 degrees or 180°/ π180°/\ \pi. This means that to convert from an angle expressed in degrees to its equivalent value in radians, all we have to do is simply multiply by 180°/ π180°/\ \pi. The degrees to radians formula is:

radians=degrees 180°π\small \text{radians} = \text{degrees} \ \cdot \cfrac{180°}{\pi}

An example where we can find radians is in the units used to express the angular velocity.

Similarly, to go from radians to degrees, multiply the angle in degrees by the conversion factor π/ 180°\pi/ \ 180°. The radians to degrees formula is as follows:

degrees=radians π180°\small \text{degrees} = \text{radians} \ \cdot \cfrac{\pi}{180°}

It's common to express polar coordinates in degrees and radians.

We gathered some notable angles in the table below. Notice that the values in radians can be expressed as multiples of pipi. Which makes it practical and an easier way to remember them:

Degrees [°]

Radians [rad]

0

0

30

π/6

45

π/4

60

π/3

90

π/2

180

π

270

3π/2

360

How to convert degrees minutes seconds to decimal degrees?

Angles are also used in disciplines other than mathematics and engineering, such as astronomy and navigation, in the form of coordinates. In these cases, it's more useful to express angles in the sexagesimal notation. This is in degrees (°), minutes ('), and seconds (") (DMS).

The equivalence here is similar to that used for measuring time. Where 1 hour equals 60 minutes and one minute equals 60 seconds. So, if we want to convert from DMS to decimal degrees, we just divide minutes by 60 and seconds by 3600 and then add it to the whole degree value.

Let's look at an example of converting DMS to decimal degrees. Suppose we want to do the angle conversion of the coordinate 33° 21' 43" into decimal degrees. We do it as follows:

  1. The whole part of the angle in degrees units remains intact. This is 33°.

  2. Next we convert the seconds into degrees by dividing by 3600:
    43" = 43"/3600 = 0.012°

  3. Now, we convert the minutes into angles by dividing by 60:
    21' = 21'/60 = 0.350°

  4. Finally, add the above conversion to the whole part:
    33° + 0.350° + 0.012° = 33.362°

This is that 33° 21' 43" is the same as 33.362°.

How to use the angle conversion calculator

With the angle conversion calculator, you can easily convert any angle value from one unit to another. You'll see that using this tool is rather simple.

All you have to do is enter the angle value in the known unit, and the converter will instantly display the angle value in many other units. As simple as that! 😄

🙋 This converter displays your results in the usual angular units such as radians or degrees and other less common units like turns (tr) or even gradians (gon).

Gabriela Diaz
deg
deg
arcmin
arcsec
rad
gon
tr
* π rad
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