Degrees to Radians Converter

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Use Degrees to Radians to convert angle measurements Degrees to Radians and vice versa.

Table of contents:

Definition of radian

The measure of an angle in radians is equal to the ratio of (the length of the arc intercepted by the angle) to (radius of the circle). Since the radian is a pure number, the "unit" [rad] cannot be written. In other words, when no angle unit is specified, the given numerical value is implicitly expressed in radians. If [rad] is sometimes added, it is to help people who are not familiar with the field. In words: "The measure of an angle in radians is equal to the length of the arc intercepted by the angle on the trigonometric circle."

Definition of Degree

The Degree is a measure of the angles of the plane which corresponds to 0-360 of a circle. Each degree can be divided into minutes and seconds. The degree has its origin in Babylon, whose numbering system was in base sixty. At that time, it was believed that the sun revolved around the Earth. Calculations indicate that it took exactly 360 days for the Sun to make a complete revolution. So, with each passing day, the Sun covered part of this path. This arc, which corresponds to the path of the Sun following its orbit for one day, was equaled by an angle, the apex of which was the center of our planet. This angle became a unit of measurement and was called a degree or angle of a degree. Although it was later shown that it was the Earth that revolved around the Sun and not the other way around, this way of measuring angles has remained until today. Another heritage of the Babylonians is the division of time into hours, minutes and seconds, which is also done on a similar scale

How to convert from degrees to radians?

To convert degrees to radians we multiply the measure of the angle by π, then we divide the result by 180 °.

Example: converting 20 ° to radians:

20 ° = 20 ° × π / 180 ° = 0.349065

How to convert from radians to degrees?

To convert radians to degrees multiply the angle measurement by 180 °, then divide the result by π.

Example: converting 0.30 to degrees:

0.30 = 0.30 × 180 ° / π = 17.18873 °

If π appears in the expression for the angle, we replace π by 180 °.

To convert degrees to radians, multiply the angle measurement by π, then divide the result by 180 °.

Why use radians?

A radian is a unit of measurement for measuring angles, such as the degree, minute of arc, grade, or thousandth. An angle of 1 radian is an angle that defines an arc of a circle with a length equal to the radius of the circle.

Let us give in a table, the values in degrees and radians of the most important angles

name of the angle value in radians value in degrees
zero angle 0 rad
milliradian 1 mrad 0°3′26″15‴
  π/6 rad 30 °
  π/4 rad 45°
  π/3 rad 60 °
radian 1 rad 57°17′44″48‴
right angle π/2 rad 90°
  2π/3 rad 120 °
  3π/4 rad 135°
flat angle π rad 180°
  5π/4 rad 225°
  3π/2 rad 270°
  7π/4 rad 315°
full angle 2π rad 360°

Small angles

For an angle of value less than 0.17 radians (ie ~ 10 °), the error is less than 1%;

For an angle of value less than 0.05 radians (i.e. ~ 3 °), the error is less than 0.1% 3.