Converting gradians to degrees is simple once the two angle systems are placed on the same scale. The key relationship is exact: 1 gradian (grad) = 0.9 degrees (deg).
This conversion is useful when a survey, mapping, engineering, or technical document uses gradians while your calculator, software, or audience expects degrees. The formula below provides a quick and reliable way to switch between them.
Gradian to Degree Conversion Factor
A full circle contains 400 gradians and 360 degrees. Therefore:
1 grad = 360 deg ÷ 400 = 0.9 deg
The exact conversion factor is:
1 grad = 0.9 deg
How to Convert Gradians to Degrees
To convert an angle from gradians to degrees, multiply the value in gradians by 0.9.
Degrees = Gradians × 0.9
- Start with the angle in gradians.
- Multiply it by 0.9.
- Label the result in degrees.
Worked Gradian to Degree Example
Suppose an angle measures 37.5 grad. Apply the conversion formula:
37.5 grad × 0.9 = 33.75 deg
Therefore, 37.5 grad equals 33.75 deg.
Whole-Number Example
For an angle of 100 grad:
100 grad × 0.9 = 90 deg
So, 100 grad equals 90 deg. This result also makes sense geometrically because 100 gradians represent one quarter of a full circle, and one quarter of 360 degrees is 90 degrees.
What Is a Gradian?
A gradian is a unit for measuring angles. It divides a full circle into 400 equal parts. In the supplied conversion notation, the symbol for gradian is grad.
Because a right angle is one quarter of a circle, it measures 100 grad. This decimal-friendly structure can be convenient for surveying and land measurement, where angles are often connected to horizontal distances and precise coordinate work.
What Is a Degree?
A degree is another unit for measuring angles. It divides a full circle into 360 equal parts. Its symbol is deg.
A right angle measures 90 deg, a straight angle measures 180 deg, and a complete revolution measures 360 deg.
Why the Conversion Works
Both gradians and degrees describe the same measure: angle. They use different divisions of a full circle, so their values are related by the ratio of their complete-circle totals:
400 grad = 360 deg
Dividing both sides by 400 gives:
1 grad = 0.9 deg
This fixed ratio applies to every valid angle measured in gradians. It does not depend on the shape, location, or application of the angle.
Common Gradian to Degree Mistakes
- Using 1.1111 instead of 0.9: The factor 1.1111... converts degrees to gradians, not gradians to degrees.
- Reversing the operation: Gradians to degrees requires multiplication by 0.9. Division by 0.9 converts degrees to gradians.
- Mixing angle units: Keep the source value in grad and the final value in deg until the calculation is complete.
- Rounding too early: Keep the full result during the calculation, then round only the final answer when needed.
Practical Applications of Gradians and Degrees
Land Surveying
Gradians are widely associated with surveying because a right angle is exactly 100 grad. This makes quarter-circle calculations and angle divisions convenient in field measurements.
Mapping and Engineering
Technical plans may use either gradians or degrees, depending on the equipment, software, or regional convention. Converting the values helps professionals compare measurements without changing the underlying angle.
Education and Technical Documentation
Degrees are more familiar in general mathematics and science, while gradians may appear in specialized instruments or survey data. A clear conversion prevents confusion when information moves between these settings.
Gradians to Degrees Conversion Table
Each row applies the same rule: multiply gradians by 0.9. Values are shown without unnecessary decimal places.
| Gradians (grad) | Degrees (deg) |
|---|---|
| 1 grad | 0.9 deg |
| 2 grad | 1.8 deg |
| 5 grad | 4.5 deg |
| 10 grad | 9 deg |
| 20 grad | 18 deg |
| 50 grad | 45 deg |
| 100 grad | 90 deg |
| 500 grad | 450 deg |
How to Convert Degrees Back to Gradians
Although the focus here is gradians to degrees, the reverse relationship can help you check your work. Since 1 grad = 0.9 deg, convert degrees to gradians by dividing by 0.9.
Gradians = Degrees ÷ 0.9
For example, 45 deg ÷ 0.9 = 50 grad. This confirms the table relationship in both directions.
Key Takeaway
The essential rule is easy to remember: multiply gradians by 0.9 to get degrees. Thus, 1 grad equals 0.9 deg, and 400 grad equals 360 deg. Keep the units visible during each step, and your angle conversions will remain clear and accurate.