Convert Degrees (°) to Gradians (grad)

What Is a Degree?

The degree (°) is a non-SI unit of plane angle defined as exactly 1/360 of a full rotation. It is accepted for use with the International System of Units due to its long-standing practical importance in navigation, surveying, engineering, astronomy, and daily measurement.

The 360-degree circle dates back to ancient Babylonian astronomy (circa 2000 BCE), likely chosen because 360 is close to the number of days in a year and is highly divisible. Today the degree remains the dominant angular unit worldwide — used in compasses, protractors, GPS coordinates, and virtually every consumer application involving angles.

Key equivalences: 1° = π/180 rad = 10/9 grad = 60 arcminutes = 3,600 arcseconds.

What Is a Gradian?

The gradian (grad), also called gon or grade, divides a right angle into exactly 100 equal parts, making a full circle equal to 400 gradians. Its exact relationship to the radian is 1 grad = π/200 rad.

Introduced during the French Revolution (1790s) as part of the metric reform effort, the gradian was designed to decimalize angle measurement — just as the meter decimalized length. While it never replaced degrees globally, it found a lasting role in continental European surveying, geodesy, and cadastral (land registry) systems, especially in France, Belgium, and Scandinavia.

Key equivalences: 100 grad = 90° = 1 right angle; 200 grad = 180°; 400 grad = 360°.

Degrees to Gradians Formula

The conversion factor is exact:

gradians = degrees × 10/9 exact

This comes from the ratio of a full circle in each system: 400 grad / 360° = 10/9 ≈ 1.11111. Equivalently, multiply degrees by 1.1̄ (1.1 repeating).

Worked Example

Convert a compass bearing of 27° to gradians:

1. Multiply: 27 × 10/9 = 270/9 = 30 grad

So 27° = exactly 30 gradians. Many degree values that are multiples of 9 convert to clean gradian numbers — a useful property when switching between systems on a surveying theodolite.

Quick Estimation

For mental math, add one-ninth to the degree value. Since 1/9 ≈ 11%, simply add 11% to the number of degrees. Example: 45° + 11% of 45 = 45 + 5 = 50 grad (exact: 50 grad). For non-round values: 72° + ~8 = ~80 grad (exact: 80 grad).

Common Conversions at a Glance

Which Is Bigger: Degree or Gradian?

A degree is 1.11 times larger than a gradian. Put another way, a gradian is 90% of a degree. The difference is small — these are the most similar-sized angular units across the two systems. A right angle is 90 degrees but 100 gradians; the gradian is simply a slightly finer subdivision of the same arc.

When You Need This Conversion

• European surveying and geodesy: Continental European surveyors — especially in France, Germany, and Scandinavia — use gradians for theodolite measurements. A bearing recorded as 27° must be entered as 30 grad when using European survey instruments.
• Scientific calculators: Most scientific calculators include a DEG/RAD/GRAD mode. Switching to grad mode is necessary when processing European survey data or verifying calculations from gradian-based references.
• Civil engineering and road design: Slope specifications in some European countries use gradians because 1 grad ≈ 1.57% slope — a nearly linear relationship that simplifies gradient calculations in road and railway design.
• Cadastral and land registry: The French cadastral system records property boundary angles in gradians. Integrating French land data with degree-based GIS systems requires this conversion.