Degrees to Gradians Converter

How to convert degrees to gradians

The two units divide the same full circle differently. Degrees split it into 360 parts; gradians split it into 400. One gradian is therefore 360/400 = 0.9°, and one degree is 400/360 = 10/9 gradians. Dividing by 0.9 and multiplying by 10/9 are the same operation, and both are exact. Going back from gradians to degrees means multiplying by 0.9, which the angle converter does alongside radians, turns and arcminutes.

The appeal of gradians is decimal neatness: a right angle is exactly 100 gon, a straight line 200 gon and a full turn 400 gon. The catch is that most degree values produce repeating decimals, since you are dividing by 9. 30° is 33.333… gon, and 1° is 1.111… gon. Multiples of 9° convert to whole numbers: 9° = 10 gon, 27° = 30 gon, 81° = 90 gon. For radians, see the degrees to radians converter.

Step by step

1. Take the angle in degrees.
2. Divide it by 0.9 (or multiply by 10 and divide by 9).
3. Round as needed; surveyors usually work to 0.0001 gon.

See it on a scale

What is a gradian?

A gradian is one four-hundredth of a full turn, exactly 0.9°, so four right angles of 100 gradians each make a circle.

The gradian was introduced in France in the 1790s, alongside the meter, as part of the push to make every measure decimal. A quarter circle became 100 grades, so a full circle is 400. The unit is known by several names: gradian, grad, grade and gon all mean the same thing, and gon is the name used in ISO 80000-3 and on surveying instruments.

Gradians never displaced degrees in everyday life, but they are still widely used in surveying and geodesy, especially in continental Europe. A total station or theodolite set to gon reads north as 0, east as 100, south as 200 and west as 300. Surveyors often subdivide into milligon (0.001 gon), and the older “centesimal minute” is one hundredth of a gon. The advantage is practical: angles add and subtract in plain decimals, with no minutes and seconds to carry.

What is GRAD mode on a calculator?

GRAD mode makes a calculator read and return angles in gradians, so sin(100) gives exactly 1 because 100 gradians is a right angle.

Many scientific calculators, including most Casio fx models, have three angle modes: DEG, RAD and GRAD, with the last one shown as a small G or GRA on the display. Graphing calculators such as the TI-84 have only degree and radian modes. If you land in GRAD mode by accident, the answers are wrong without looking obviously wrong:

• sin(90) returns 0.9877 instead of 1, because the calculator reads 90 gon = 81°.
• tan(45) returns 0.8541 instead of 1, since 45 gon is only 40.5°.

Some calculators also have a DRG or DRG▸ key that converts a value between modes rather than just switching the mode. The same checks as the radians to degrees page apply: know what sin(30) should be, and test it.

Mils, hours and other ways to divide a circle

Gradians are one of several alternatives to the 360-degree circle:

• NATO mil: 6,400 per circle, used for military aiming. One gon is exactly 16 NATO mils, and one degree is 17.7778 mils.
• Hour angle: astronomers split the circle into 24 hours of right ascension, so 1 hour = 15° = 16.6667 gon.
• Turns: engineers sometimes simply count revolutions; one turn is 360°, 400 gon or 2π rad.

Whatever the unit, the conversion is always a ratio of how many parts each system puts in a full circle: 360 degrees, 400 gradians, 6,400 NATO mils or 2π radians.