Converting degrees to gradians is straightforward once the circle-based relationship is clear. The key fact is that a full angle contains 360 degrees (deg) or 400 gradians (grad).
Exact conversion factor: 1 degree = 10/9 gradians, or approximately 1.111111 gradians.
Degree to Gradian Formula
To convert an angle from degrees to gradians, multiply the degree value by 10/9:
Gradians = Degrees × 10/9
For a decimal result, the same relationship can be written as:
Gradians = Degrees × 1.111111…
The repeating decimal represents the exact fraction 10/9. For accurate work, especially in surveying and engineering, keeping the fraction until the final step can reduce rounding errors.
What Is a Degree?
A degree is a unit of angle measurement. A complete turn is divided into 360 equal parts, so one degree represents 1/360 of a full circle.
Degrees are commonly used in geometry, navigation, mapping, construction, education, and everyday descriptions of direction. For example, a right angle measures 90 degrees, while a straight angle measures 180 degrees.
What Is a Gradian?
A gradian is also a unit of angle within the Angle measure. A complete turn contains 400 gradians, making a right angle equal to 100 gradians.
Because 400 divides conveniently into four right angles, gradians are useful in land surveying, civil engineering, and some technical mapping systems. The supplied symbol for the unit is grad.
How Degrees and Gradians Are Related
Both units describe the same geometric quantity: angle. They use different divisions of a complete circle.
- Full circle: 360 deg = 400 grad
- Right angle: 90 deg = 100 grad
- Straight angle: 180 deg = 200 grad
- Half of a right angle: 45 deg = 50 grad
Since the same angle has a larger numerical value in gradians than in degrees, a degree value increases by a factor of 10/9 when converted to gradians.
Worked Example: Convert 45 Degrees to Gradians
Start with the formula:
Gradians = Degrees × 10/9
Insert 45 degrees:
45 deg × 10/9 = 450/9 = 50 grad
Therefore, 45 deg equals 50 grad. This result is exact because 45 degrees is half of a right angle, and a right angle contains 100 gradians.
More Degree to Gradian Examples
Convert 30 Degrees
30 deg × 10/9 = 33.3333 grad
So, 30 degrees is approximately 33.3333 gradians.
Convert 90 Degrees
90 deg × 10/9 = 100 grad
So, a right angle measures exactly 100 gradians.
Convert 180 Degrees
180 deg × 10/9 = 200 grad
So, a straight angle measures exactly 200 gradians.
When Degree to Gradian Conversion Is Useful
Most general-purpose angle measurements use degrees. However, gradians can be more convenient in technical fields because a right angle divides evenly into 100 parts.
- Land surveying: Surveyors may use gradians to express horizontal and vertical angles.
- Civil engineering: Road grades, terrain measurements, and construction layouts may involve gradian-based instruments or plans.
- Mapping: Converting between angle systems helps when data comes from different surveying standards.
- Education: The relationship provides a practical example of equivalent units within the Angle measure.
- Technical instruments: Some theodolites and surveying devices display angles in gradians rather than degrees.
Tips for Accurate Conversion
- Confirm that the starting value is expressed in degrees, using the symbol deg.
- Multiply the value by 10/9, not by 9/10. The reverse factor applies when converting gradians to degrees.
- Keep the fraction 10/9 during the calculation when precision matters.
- Round only the final result unless the required precision is low.
- Check familiar reference angles, such as 90 deg = 100 grad, to catch input or calculation errors.
Degrees to Gradians Conversion Table
Each row applies the same rule: multiply degrees by 10/9. Decimal results are rounded to 4 decimal places where needed.
| Degree (deg) | Gradian (grad) |
|---|---|
| 1 deg | 1.1111 grad |
| 2 deg | 2.2222 grad |
| 5 deg | 5.5556 grad |
| 10 deg | 11.1111 grad |
| 20 deg | 22.2222 grad |
| 50 deg | 55.5556 grad |
| 100 deg | 111.1111 grad |
| 500 deg | 555.5556 grad |
Reverse Conversion: Gradians to Degrees
To convert gradians back to degrees, use the inverse relationship:
Degrees = Gradians × 9/10
For example, 200 gradians becomes:
200 grad × 9/10 = 180 deg
This confirms the central relationship: 360 deg = 400 grad.
Key Takeaway
Degree to gradian conversion uses one exact rule: multiply the degree value by 10/9. A full circle is 360 deg or 400 grad, and a right angle is 90 deg or 100 grad. Apply the formula, retain extra digits when precision matters, and round the final result to suit the task.