About the Angle Converter

The angle converter translates between degrees, radians, gradians, turns, arcminutes, and arcseconds. Whether you are working on a geometry problem in degrees, a calculus problem in radians, a surveying calculation in gradians, or an astronomy measurement in arcseconds, this tool gives you an instant conversion in either direction.

The degree, dating back to ancient Babylon, divides a full circle into 360 parts. The radian, the SI derived unit of angle, defines a full circle as 2 times pi radians (about 6.283). The gradian (or gon) divides a full circle into 400 parts, which makes each quadrant a round 100 gradians. The turn (or revolution) is simply one full circle. The arcminute and arcsecond are subdivisions of a degree used in astronomy and navigation.

The choice of unit depends on the field. Trigonometry and calculus use radians because the derivatives of sine and cosine take their simplest form in radians. Engineering and navigation use degrees. Surveying uses gradians in some European countries. Astronomy uses arcseconds because the angles involved are extremely small.

How It Works

Each unit has a defined relationship to the degree. The converter uses these factors to translate between any two units:

1 deg    = 1 degree
1 rad    = 57.2957795 deg   (180 / pi)
1 grad   = 0.9 deg          (1/400 of a full circle)
1 turn   = 360 deg          (one full revolution)
1 arcmin = 0.0166667 deg    (1/60 of a degree)
1 arcsec = 0.000277778 deg  (1/3600 of a degree)

Every conversion proceeds in two steps: the input is multiplied by the source unit's factor to give an angle in degrees, then divided by the target unit's factor to give the result in the target unit. The factor for radians is 180 divided by pi, since a full circle is 2 times pi radians or 360 degrees.

The relationships between the units follow from the way each divides the full circle. A degree is 1/360 of a circle, a radian is 1/(2 pi) of a circle, a gradian is 1/400 of a circle, a turn is the whole circle, an arcminute is 1/21600 of a circle, and an arcsecond is 1/1296000 of a circle. These divisions reflect historical conventions rather than any intrinsic mathematical relationship.

Worked Examples

To convert 90 degrees to radians, the calculation is 90 / 57.2957795 = 1.5708 radians, which is pi/2. This is a right angle, and the value pi/2 is one of the most commonly encountered angles in mathematics and engineering.

To convert 1 radian to degrees, the calculation is 1 * 57.2957795 = 57.2958 degrees. This is the angle subtended by an arc equal in length to the radius, and it is the natural unit of angle measurement in calculus because the derivative of sin(x) is cos(x) only when x is in radians.

To convert 1 arcsecond to degrees, the calculation is 1 * 0.000277778 = 0.000277778 degrees. This is an extremely small angle, about the apparent size of a coin viewed from a distance of several kilometres. Astronomers use arcseconds because the angles subtended by stars and planets are tiny.

When to Use This Tool

Use the angle converter whenever you need to translate between angle units:

Converting between degrees and radians for trigonometry and calculus.
Interpreting astronomical measurements in arcseconds or arcminutes.
Working with surveying data in gradians (used in some European countries).
Converting compass bearings between degrees and turns.
Translating engineering specifications between metric and imperial angle units.
Working with geographic coordinates expressed in degrees, minutes, and seconds.
Teaching the relationships between the various angle units.

Limitations & Disclaimer

This converter handles plane angles only and does not convert solid angles (steradians), astronomical right ascension (hours, minutes, seconds), or compass bearings with quadrant indicators. The conversion to and from radians is limited by the precision of pi. For astronomical or geodetic work requiring sub-arcsecond precision, use dedicated software. See our disclaimer for full details.

Frequently Asked Questions

Why are there 360 degrees in a circle?

The 360-degree division dates back to ancient Babylon, which used a base-60 number system. The number 360 was chosen because it is divisible by many small integers (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180), which makes it convenient for dividing a circle into equal parts.

What is a radian and why is it the SI unit?

A radian is the angle subtended by an arc whose length equals the radius of the circle. A full circle is 2 times pi radians. It is the SI unit because the derivatives of trigonometric functions take their simplest form in radians: the derivative of sin(x) with respect to x is cos(x) only when x is measured in radians.

What is a gradian and where is it used?

A gradian (or gon) is 1/400 of a full circle, so a right angle is exactly 100 gradians. It was introduced during the French Revolution as part of the metric system and is still used in surveying in some European countries, particularly France and Germany. It makes calculations involving right angles cleaner but never displaced the degree in general use.

What are arcminutes and arcseconds used for?

They are subdivisions of a degree: one arcminute is 1/60 of a degree, and one arcsecond is 1/3600 of a degree. They are used in astronomy (where the angles subtended by stars are tiny), in navigation (for expressing latitude and longitude), and in optics (for measuring angular resolution). One arcsecond is about the apparent size of a coin viewed from kilometres away.

How do I convert degrees, minutes, and seconds to decimal degrees?

Divide the minutes by 60 and the seconds by 3600, then add them to the degrees. For example, 45 degrees, 30 minutes, 15 seconds equals 45 + 30/60 + 15/3600 = 45.5042 degrees. The converter handles decimal degrees directly; convert DMS to decimal first if needed.

Are the conversions exact?

The conversions between degrees, gradians, turns, arcminutes, and arcseconds are exact because they are defined as simple fractions of a full circle. The conversion to and from radians involves pi, which is irrational, so the factor 180/pi is approximate. The converter uses a high-precision value of pi, giving about 15 significant digits of accuracy.

Last updated: July 21, 2026 · Author: HT99 Tools Editorial Team · Reviewed by: HT99 Tools Editorial Team