How to Convert Degrees to Gradians
Converting degrees to gradians (also called gons or grads) is a necessary operation in land surveying, civil engineering, and certain European technical traditions. The gradian divides a right angle into 100 equal parts, making a full circle 400 gradians instead of the 360 degrees used in the degree system. This decimal-friendly structure was introduced during the French Revolution as part of the broader effort to decimalize all measurement systems. Today, gradians remain the standard angular unit in surveying and cartography in many European countries, including France, Germany, and Scandinavia. Surveying instruments manufactured in Europe often display readings in gradians, and topographic maps in these regions use gradian-based grid systems. When engineers and surveyors trained in the degree system need to work with European survey data, equipment, or maps, this conversion becomes essential. The relationship between degrees and gradians is straightforward and based on a simple ratio, making it easy to learn and apply in practical fieldwork and office calculations alike.
Conversion Formula
The conversion from degrees to gradians uses the factor 10/9 (approximately 1.11111). This factor comes from the relationship between the two systems: a right angle is 90 degrees or 100 gradians. The ratio 100/90 simplifies to 10/9. Since gradians divide the circle into finer increments (400 versus 360), the numerical value in gradians is always larger than in degrees. This clean ratio reflects the fact that both 360 and 400 are multiples of common factors, enabling a simple fractional conversion.
grad = deg × 10/9
5 degrees = 5.55556 gradians
Step-by-Step Example
To convert 5 degrees to gradians:
1. Start with the value: 5 degrees
2. Multiply by the conversion factor: 5 × 10/9
3. Calculate: 5 × 1.11111 = 5.55556
4. Result: 5° = 5.55556 gradians
Understanding Degrees and Gradians
What is a Degree?
The degree has been used as a unit of angular measurement since ancient Babylon, approximately 3000 BCE. The Babylonian sexagesimal system naturally divided the full circle into 360 parts. Greek astronomers adopted this system, and Claudius Ptolemy's influential Almagest (circa 150 CE) cemented the 360-degree circle in Western mathematical and astronomical tradition. The degree has been subdivided into 60 arcminutes and 3,600 arcseconds, following the same base-60 convention, and remains the most universally recognized angular unit in navigation, geography, and everyday use.
What is a Gradian?
The gradian was introduced in France during the French Revolution as part of the metric system of measurements, first proposed in 1791. The revolutionary government sought to decimalize all units, and the gradian divided the right angle into 100 parts instead of the traditional 90 degrees. The unit was initially called the "grade" and was incorporated into French surveying and mapping standards. Although the broader attempt to decimalize time failed, the gradian persisted in surveying and civil engineering, where its decimal structure simplifies calculations. Today, the International System of Units lists the gradian as an accepted non-SI unit.
Practical Applications
This conversion is heavily used in European land surveying, where total station instruments and theodolites may be calibrated in gradians. French and German topographic maps use gradian-based coordinate grids, so engineers working with these maps must convert degree-based GPS readings to gradians. Military artillery calculations in some European nations use gradians (in the form of milliradians, where 1 gradian = 10 milliradians in the NATO system) for fire direction. Civil engineers designing roads, tunnels, and railways in gradian-using countries perform this conversion when integrating data from degree-based sources such as satellite positioning systems.
Tips and Common Mistakes
A common error is using 9/10 instead of 10/9, which converts in the wrong direction (gradians to degrees). Remember that a gradian is smaller than a degree, so the numerical value in gradians should be larger. Another mistake is confusing gradians with radians; these are completely different units. As a quick check, 90° should equal exactly 100 gradians, 180° should equal 200 gradians, and 360° should equal 400 gradians. If your result does not scale proportionally with these reference points, verify your calculation.
Frequently Asked Questions
A gradian (also called a gon or grad) is a unit of angle equal to 1/400 of a full circle. A right angle is exactly 100 gradians. The gradian system was created to provide a decimal-based alternative to degrees, making arithmetic with angles simpler in surveying and engineering applications.