About the Angle Converter

The angle converter translates between six units of angle: degree, radian, gradian, turn, arcminute, and arcsecond. All conversions are expressed in degrees as the pivot unit, with the radian defined by the exact ratio 180/pi.

The degree divides a full turn into 360 parts, a convention that originated with the Babylonians around 2000 BCE. The choice of 360 was likely motivated by its many divisors (24 in total) and its approximate match to the number of days in a year. The degree remains the standard unit for everyday angle measurement in geometry, navigation, and engineering.

The radian is the SI derived unit of angle, defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. One full turn equals 2*pi radians, so one radian equals 180/pi degrees (approximately 57.2958°). Radians are used in calculus and physics because they simplify the derivatives of trigonometric functions: d/dx sin(x) = cos(x) only when x is in radians.

The gradian divides a full turn into 400 parts, with 100 gradians in a right angle. It was introduced during the French Revolution as part of the metric system, intended to simplify surveying (100 grad = 90°). The turn is the full circle, equal to 360 degrees. The arcminute and arcsecond are subdivisions of the degree: 1 degree = 60 arcminutes = 3600 arcseconds. These small units are used in astronomy and optics.

How It Works

The converter uses the two-step method through degrees:

base_in_deg = value * factor[from]
result = base_in_deg / factor[to]

The factor table (in degrees) is:

deg   = 1°              rad = 57.2958° (= 180/pi)
grad  = 0.9° (= 1/400 turn * 360°/turn)
turn  = 360°           arcmin = 1/60°  arcsec = 1/3600°

The factor for radians is exactly 180/pi, which is approximately 57.29577951. The displayed value 57.2958 is rounded; the actual computation uses the full-precision value via Math.PI (which itself is precise to about 16 digits).

The gradian is exactly 0.9 degrees because 400 gradians equal one full turn (360 degrees), so 1 gradian = 360/400 = 0.9 degrees. The arcminute is 1/60 of a degree and the arcsecond is 1/3600 of a degree, following the sexagesimal (base-60) system inherited from the Babylonians.

The two-step method through degrees keeps the code simple. The pivot unit also matches the everyday convention: angle measurements in degrees are intuitive, while radians and gradians are more specialized.

Worked Examples

Converting 90 degrees to radians: the input in degrees is 90, and the result is 90 / 57.2958 = 1.5708 rad. This equals pi/2, which is the angle of a right angle in radians. The displayed value 1.5708 is rounded; the actual computation gives 1.5707963.

Converting 1 radian to degrees: the input in degrees is 57.2958, displayed directly. This is the angle subtended by an arc equal in length to the radius, which is approximately 57.3 degrees. A full circle is 2*pi radians, or about 6.2832 radians.

Converting 100 gradians to degrees: the input in degrees is 100 * 0.9 = 90 degrees. This is the right angle that the gradian was designed to make round-number: 100 grad = 90°, 200 grad = 180°, 300 grad = 270°, 400 grad = 360°.

When to Use This Tool

Use the angle converter when you need to:

Convert trigonometric inputs between degrees and radians for scientific computing.
Translate surveying measurements between gradians and degrees.
Convert astronomical angles from arcseconds to degrees for celestial navigation.
Convert machinist angles from degrees to radians for CNC programming.
Translate compass headings from degrees to radians for navigation software.
Convert camera field-of-view specifications between degrees and radians.
Verify the result of an inverse trigonometric function (which returns radians).

Limitations & Disclaimer

This converter uses the standard conversion factors between degrees, radians, gradians, turns, arcminutes, and arcseconds. It does not handle solid angles (steradians), the historical "point" used in navigation (11.25 degrees), or the milliradian used in ballistics. Results are subject to IEEE 754 floating-point precision. The factor for radians (180/pi) is irrational, so conversions involving radians are approximate. See our disclaimer for full details.

Frequently Asked Questions

Why are there 360 degrees in a full turn?

Because the Babylonians used a sexagesimal (base-60) number system and observed that the sun moves about 1/360 of a circle per day. The number 360 has 24 divisors (including 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 60, 90, 120, 180), which makes it convenient for dividing into equal parts.

Why is the radian the SI unit of angle?

Because it is defined purely in terms of length ratios, without an arbitrary subdivision. The radian is the angle subtended by an arc equal in length to the radius, so it is a dimensionless ratio of two lengths. Per the SI Brochure 9th edition, the radian is a derived unit of dimensionless quantity, equal to 1.

What is the gradian used for today?

The gradian (also called the gon) is used in some European surveying and construction applications, particularly in France and Germany. It was introduced during the French Revolution as part of the metric system, with 100 gradians in a right angle. Most other fields use degrees or radians.

What is the difference between an arcminute and an arcsecond?

An arcminute is 1/60 of a degree, and an arcsecond is 1/60 of an arcminute, or 1/3600 of a degree. These small units are used in astronomy (the angular resolution of a telescope is often quoted in arcseconds), in optics (visual acuity is measured in arcminutes), and in surveying.

How does the converter handle negative angles?

It accepts them normally. Negative angles represent clockwise rotation, while positive angles represent counterclockwise rotation (by the standard mathematical convention). The conversion factor is the same regardless of sign, so a negative input produces a negative result of the same magnitude.

What is a steradian and how does it differ from a radian?

A steradian is the SI unit of solid angle, the three-dimensional analogue of the radian. A full sphere subtends 4*pi steradians. The radian measures planar (two-dimensional) angle, while the steradian measures solid (three-dimensional) angle. This converter handles only planar angles; for solid angles, use a dedicated steradian converter.

Last updated: September 9, 2026 · Author: HT99 Tools Editorial Team