Degrees to Gradians Converter

Convert Degrees to Gradians Online

Converting degrees to gradians is a practical task for surveyors, civil engineers, and anyone working with metric angle measurement systems. Our free online converter lets you instantly translate any degree value into gradians with precision. Simply enter your angle in degrees and receive the gradian equivalent without manual calculation or lookup tables.

Degrees to Gradians Conversion Formula

The relationship between degrees and gradians is based on how each system divides a full circle. A complete rotation equals 360 degrees and also equals 400 gradians. This clean ratio makes the conversion formula simple and easy to remember, with no irrational numbers like pi involved.

The Core Formula

To convert any angle from degrees to gradians, use the following equation:

gradians = degrees x (400 / 360)

This simplifies to:

gradians = degrees x (10 / 9)

Or equivalently:

gradians = degrees x 1.1111

The exact conversion factor is 10/9, which is a repeating decimal 1.1111... This means that every degree equals approximately 1.1111 gradians. The factor comes directly from dividing the gradian full circle (400) by the degree full circle (360), giving 400/360 which reduces to 10/9.

Step-by-Step Conversion Process

Follow these steps to convert any degree value to gradians manually:

Step 1: Take your angle in degrees. For example, let us convert 90 degrees to gradians.

Step 2: Multiply the degree value by 10. So 90 x 10 = 900.

Step 3: Divide the result by 9. So 900 / 9 = 100.

Step 4: The answer is 100 gradians. A right angle is exactly 100 gradians, which is one of the key advantages of the gradian system.

Let us try another example. Convert 45 degrees to gradians:

Step 1: Start with 45 degrees.

Step 2: Multiply by 10: 45 x 10 = 450.

Step 3: Divide by 9: 450 / 9 = 50.

Step 4: The result is 50 gradians.

For a non-standard angle, convert 75 degrees: 75 x 10 / 9 = 83.333 gradians. Notice that values which are not multiples of 9 produce repeating decimals in gradians.

Why the Formula Works

The gradian, also known as the gon or grade, was designed as part of the metric system during the French Revolution. The creators wanted a decimal-friendly angle unit where a right angle would be exactly 100 units, making calculations with percentages and decimal fractions more intuitive. A full circle became 400 gradians, a right angle 100 gradians, and a straight angle 200 gradians. Since the degree system uses 360 for a full circle, the ratio between the two systems is 400 to 360, which simplifies to 10 to 9. This rational ratio means conversions between degrees and gradians never involve irrational numbers, unlike conversions involving radians.

About Degrees and Gradians

Understanding the Conversion

The degree is the most widely recognized angle unit in everyday life. It divides a full circle into 360 parts, a convention inherited from Babylonian astronomy over two thousand years ago. Degrees are used in navigation, weather reports, architecture, and most educational curricula worldwide. Their familiarity makes them the default choice for communicating angles to a general audience.

The gradian was introduced in France during the late 18th century as part of the broader push toward metrication. The idea was to create an angle unit compatible with the decimal system. By setting a right angle to exactly 100 gradians, calculations involving slopes, grades, and percentages become more straightforward. A one-percent slope, for instance, corresponds closely to one gradian of inclination at small angles, which is a convenient relationship for road and railway engineering.

Today, gradians are most commonly used in surveying, land measurement, and civil engineering, particularly in continental Europe and parts of the Middle East. Many surveying instruments and total stations offer a gradian mode alongside degrees. If you need to convert between radians and degrees instead, our radians to degrees converter is available for that purpose.

Practical Applications

Converting degrees to gradians is essential in several professional fields where the gradian system offers practical advantages:

Land Surveying: Surveyors in many European countries use gradians as their primary angle unit. When survey data from international projects arrives in degrees, it must be converted to gradians for compatibility with local instruments and records. The decimal nature of gradians simplifies field calculations, especially when dividing angles into equal parts.

Road and Railway Engineering: Slope gradients in civil engineering are often expressed as percentages. At small angles, the tangent of an angle in gradians is very close to the angle value divided by 100, making gradians a natural fit for slope calculations. A road with a 5-percent grade corresponds to approximately 5 gradians of inclination.

Mining and Tunneling: Underground surveying frequently uses gradians because the decimal subdivision makes it easier to work with angular measurements in confined spaces where quick mental arithmetic is valuable. Tunnel boring machines and mining equipment in European operations often display bearings in gradians.

Military Applications: Some military organizations use the mil, which is closely related to the gradian. The NATO mil divides a circle into 6400 parts, while the gradian divides it into 400. Converting between degrees and gradians is sometimes a stepping stone in military angle calculations.

For a complete set of angle conversion tools, our angle converter tool supports degrees, radians, gradians, and other angle units in one interface.

Quick Tips

Here are some helpful tricks for working with degrees to gradians conversions:

Multiply by 10 and Divide by 9: This is the simplest way to remember the conversion. For multiples of 9 degrees, the result is always a whole number of gradians. For example, 9 degrees equals 10 gradians, 18 degrees equals 20 gradians, and 27 degrees equals 30 gradians.

Right Angle Benchmark: Remember that 90 degrees equals exactly 100 gradians. This is the easiest reference point. Half a right angle (45 degrees) is 50 gradians. A straight angle (180 degrees) is 200 gradians. A full circle (360 degrees) is 400 gradians.

Add About 11 Percent: Since the conversion factor is 10/9 or about 1.111, the gradian value is always roughly 11 percent larger than the degree value. This gives you a quick sanity check for your calculations.

Reverse Conversion Shortcut: To go from gradians back to degrees, multiply by 9 and divide by 10, or equivalently multiply by 0.9. So 100 gradians times 0.9 equals 90 degrees. This inverse relationship is easy to remember.

For converting between degrees and the mathematically standard radian unit, check out our degrees to radians converter for instant results.

Degrees to Gradians Reference Table

Degrees	Gradians
0	0
9	10
18	20
30	33.33
45	50
60	66.67
72	80
90	100
120	133.33
135	150
150	166.67
180	200
225	250
270	300
315	350
360	400

Frequently Asked Questions

What is the formula for converting degrees to gradians?

The formula is gradians = degrees x (10 / 9). You multiply the degree value by 10 and then divide by 9. This works because a full circle is 360 degrees or 400 gradians, and the ratio 400/360 simplifies to 10/9. For example, 90 degrees times 10/9 equals 100 gradians. The conversion factor as a decimal is approximately 1.1111.

What is a gradian?

A gradian, also called a gon or grade, is a unit of angle measurement where a full circle equals 400 gradians. A right angle is exactly 100 gradians, a straight angle is 200 gradians, and a three-quarter turn is 300 gradians. The gradian was created during the French Revolution as part of the metric system to provide a decimal-friendly alternative to degrees. It is primarily used in surveying and civil engineering.

How many gradians are in a right angle?

A right angle contains exactly 100 gradians. This is one of the primary advantages of the gradian system. In degrees, a right angle is 90, and in radians it is pi/2 (approximately 1.5708). The clean value of 100 gradians for a right angle makes decimal subdivision straightforward, which is why surveyors and engineers find gradians convenient for field calculations.

Who uses gradians?

Gradians are primarily used by land surveyors, civil engineers, and geodesists, especially in France, Germany, Switzerland, and other European countries. Many total stations and theodolites used in surveying offer a gradian mode. Some scientific calculators also include a gradian setting alongside degrees and radians. In everyday life, gradians are rarely encountered outside these professional contexts.

What is the difference between gradians and degrees?

The main difference is how they divide a full circle. Degrees split a circle into 360 equal parts, while gradians split it into 400 equal parts. This means one gradian equals 0.9 degrees, and one degree equals approximately 1.1111 gradians. Degrees have ancient Babylonian origins and are universally recognized. Gradians are a modern metric invention designed for decimal convenience, primarily used in surveying and engineering.

Are gradians the same as radians?

No, gradians and radians are completely different units. Radians are based on the mathematical relationship between arc length and radius, with a full circle equaling 2pi (approximately 6.2832) radians. Gradians divide a full circle into exactly 400 equal parts. One radian equals approximately 63.662 gradians. The two systems serve different purposes: radians are preferred in pure mathematics and physics, while gradians are used in surveying and metric engineering.

How do I convert gradians back to degrees?

To convert gradians to degrees, multiply the gradian value by 9/10 or equivalently by 0.9. For example, 200 gradians times 0.9 equals 180 degrees. This is simply the inverse of the degrees-to-gradians formula. You can also think of it as multiplying by 360 and dividing by 400, which reduces to the same 9/10 ratio.

Can my calculator work in gradians?

Most scientific calculators have a mode setting that lets you switch between degrees, radians, and gradians. Look for a button labeled DRG or a mode menu that shows DEG, RAD, and GRAD options. When set to gradian mode, trigonometric functions will accept and return values in gradians. Make sure to check your calculator mode before performing calculations, as using the wrong angle unit is a common source of errors.