Angle Converter (degrees, DMS, radians, gradians)

Understanding the angle converter — concept and principle

Angles are written in several systems: decimal degrees in GPS apps and spreadsheets, degrees–minutes–seconds (DMS) on maps and in surveying, radians in mathematics and programming, gradians (gon) in some surveying instruments, and turns in engineering and graphics. This converter accepts an angle in any of these forms and returns all the others at once, which is handy for converting a map coordinate, checking a trigonometry answer or setting up a CAD drawing.

How it is calculated

Everything is first converted to decimal degrees d, then:

• DMS: D = whole degrees of |d|; M = whole minutes of (|d| − D) × 60; S = ((|d| − D) × 60 − M) × 60, shown to three decimals.
• Radians = d × π ÷ 180
• Gradians = d ÷ 0.9 (400 gon = 360°)
• Turns = d ÷ 360
• Arcminutes = d × 60; arcseconds = d × 3,600

When you enter DMS, the calculator uses d = D + M ÷ 60 + S ÷ 3,600, with the sign of the degrees field applied to the whole angle.

The principle

A full circle is divided differently by each system. Degrees split it into 360, a Babylonian heritage in base 60, which is why each degree has 60 minutes and each minute 60 seconds. Gradians split the circle into 400 so that a right angle is a round 100. The radian is the natural unit: an angle of 1 radian cuts off an arc equal to the radius, so a full circle is 2π radians. Because all of them measure the same thing, converting is simply multiplying by the ratio of their circle sizes.

Worked example

1. Input: 37.5665° (a latitude often used for central Seoul).
2. D = 37; remaining 0.5665° × 60 = 33.99′, so M = 33.
3. S = (33.99 − 33) × 60 = 59.400″, giving 37° 33′ 59.400″.
4. Radians = 37.5665 × π ÷ 180 = 0.6556591135.
5. Gradians = 37.5665 ÷ 0.9 = 41.74055556; turns = 0.1043513889.
6. Arcminutes = 2,253.99; arcseconds = 135,239.4.

How to read the result

For coordinates, one second of latitude is roughly 31 m on the ground and one minute about 1.85 km (one nautical mile), so the number of decimals tells you the precision: four decimals of a degree locate a point to about 11 m. Radians are what programming functions such as sin and cos expect. Negative angles mean south latitude, west longitude or clockwise rotation, depending on the convention.

Common mistakes

• Typing 37.33 when you mean 37° 33′; decimal degrees are not minutes.
• Passing degrees to a sin() or cos() function that expects radians.
• Losing the minus sign on a southern or western coordinate when converting DMS.
• Confusing gradians with degrees on a calculator set to GRAD mode.

Frequently asked questions

How many degrees is one radian?

About 57.29577951°, which is 180 ÷ π.

How do I convert DMS to decimal degrees?

Add degrees + minutes ÷ 60 + seconds ÷ 3,600, for example 37 + 33/60 + 59.4/3,600 = 37.5665°.

Where are gradians used?

Mainly in surveying in some European countries, where a right angle of 100 gon makes slope and bearing calculations simpler.

Sources: BIPM, The International System of Units (SI Brochure), 9th edition · NIST Special Publication 811 (2008), Guide for the Use of the International System of Units, Appendix B