Angle Converter
180 degrees equals pi radians, and 200 gradians describe the same half turn. This angle converter changes an angle value between degrees, radians, and gradians without making you recalculate each relationship by hand. Enter the number you have, choose the source unit, choose the target unit, and the tool converts through a degree base so the result stays consistent across all three systems. That is useful when a geometry problem is written in degrees, a trigonometry formula expects radians, or a surveying note uses gradians instead of degrees. The page also helps with quick landmark checks. If your answer should match a quarter turn, half turn, or full turn, you can compare the converted result with 90 degrees, 180 degrees, or 360 degrees before using it in the next step of your work.
Quick answer
A full circle equals 360 degrees, 2 pi radians, or 400 gradians, so each unit system describes the same turn with a different scale.
Result
3.141593
What this tells you
- •A full circle equals 360 degrees, 2 pi radians, or 400 gradians, so each unit system describes the same turn with a different scale.
- •Radians connect angle size to arc length, which is why trigonometry, precalculus, and calculus often prefer them.
- •Gradians split a right angle into 100 parts, which makes them convenient for decimal-based surveying and some engineering references.
- •Landmark checks help you catch mistakes fast. 90 degrees = pi/2 radians = 100 gradians, and 180 degrees = pi radians = 200 gradians.
- •Negative angles and decimal values convert the same way as positive whole numbers because the unit relationships stay fixed.
- •The displayed result rounds to 6 decimal places, which is usually enough for schoolwork, graphing, and everyday planning.
How to Use
- 11. Enter the angle value you want to convert. Whole numbers, decimals, and negative values all work.
- 22. Choose the source unit in the From field. Pick degrees, radians, or gradians based on the number you already have.
- 33. Choose the target unit in the To field. The tool uses the same base conversion method for every pair of units.
- 44. Read the converted angle in the result area and compare it with a known benchmark when possible, such as a right angle or straight angle.
- 55. Use the converted value in your equation, drawing, graph, or measurement note, keeping in mind that the tool shows decimal output rather than symbolic pi.
How It Works
Formula
degrees = value x source factor, result = degrees / target factorThe tool uses degrees as the shared base unit. It first multiplies your input by a source factor to turn the starting value into degrees. Those factors match the formula code exactly: degrees = 1, radians = 180/pi, and gradians = 0.9. After that, it divides by the factor for the target unit. Example: 2 radians x 180/pi = 114.591559 degrees. If you then want gradians, divide 114.591559 by 0.9 and you get 127.323954 gradians. The familiar one-step formulas still fit inside this method. Degrees to radians multiplies by pi/180, radians to degrees multiplies by 180/pi, and degrees to gradians multiplies by 10/9 because 400 gradians and 360 degrees represent the same full circle. Using one degree-based path keeps direct conversions like radians to gradians consistent with the simpler degree-based formulas you may already know.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Convert 45 degrees to radians
Multiply 45 by pi/180 to get pi/4, which the tool displays as 0.785398 after rounding. This is a standard unit-circle angle that appears often in geometry and trigonometry. Because 45 degrees is half of a right angle, the radian result should also be half of pi/2.
Convert 1 radian to degrees
Multiply 1 by 180/pi to convert the radian measure into degrees. The result is 57.29578 degrees after rounding, which shows that one radian is a little more than half of 90 degrees. This benchmark helps when you want to picture a radian without drawing the circle.
Convert 100 gradians to radians
One hundred gradians is a right angle because 400 gradians make a full circle. A right angle is also 90 degrees, and 90 degrees equals pi/2 radians, which appears here as 1.570796 after rounding. This is a strong cross-check because all three unit systems land on the same quarter turn.
Convert 225 degrees to gradians
Since 200 gradians equals 180 degrees, each degree equals 10/9 gradians. Multiplying 225 by 10/9 gives 250 gradians exactly. This kind of conversion is useful when an angle written in degrees needs to match a gradian-based drawing or field note.
Convert 2.5 radians to gradians
First convert 2.5 radians to degrees by multiplying by 180/pi, which gives 143.239449 degrees in the working step. Then divide by 0.9 to change degrees into gradians, which gives 159.154943. This example shows why the degree-base method is useful when you want a direct radians-to-gradians answer.
Convert 75 gradians to degrees
One gradian equals 0.9 degree, so 75 x 0.9 = 67.5 degrees. The result is smaller than 75 because a gradian is slightly less than a degree. This is a simple sense check when you want to confirm that the unit change moved in the expected direction.
Common Angle Conversions
Use these benchmark angles to sense-check a conversion before you plug the result into another formula.
| Degrees | Radians | Gradians |
|---|---|---|
| 30 | pi/6 (0.5236) | 33.33 |
| 45 | pi/4 (0.7854) | 50 |
| 60 | pi/3 (1.0472) | 66.67 |
| 90 | pi/2 (1.5708) | 100 |
| 180 | pi (3.1416) | 200 |
| 360 | 2 pi (6.2832) | 400 |
Radian values are shown as common multiples of pi plus decimals rounded to 4 places. The 30-degree and 60-degree gradian values are rounded repeating decimals.
Common mistakes
- Entering a radian value while the From field still says degrees. The numbers look similar on the page, but 1 degree and 1 radian are very different angles.
- Forgetting that the tool shows decimal output instead of symbolic pi. A result like 3.141593 means pi radians, not a new angle.
- Mixing gradians with degrees on right-angle problems. A right angle is 90 degrees but 100 gradians.
- Rounding too early when you plan to reuse the converted angle in another calculation. Keep more digits until the last step if precision matters.
- Expecting the same number to carry across units. The angle size stays the same, but the numeric value changes because each system uses a different scale.
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