Angle Converter (All-in-One)
Convert between degrees, radians, and grads instantly
Converted Values
Degree (°): 90.00
Radian (rad): 1.57
Gradian (gon): 100.00
Formula
Radian = Degree × (π / 180)
Degree = Radian × (180 / π)
Gradian = Degree × (10 / 9)
What is it?
This angle converter converts between degrees, radians, and gradians simultaneously — widely used in mathematics, trigonometry, physics, and engineering.
Formula
- 180° = π radians
- 90° = π/2 radians
- 1 radian ≈ 57.2958°
Formula Explanation
A full circle spans 360 degrees, 2π radians, or 400 gradians — three different ways of dividing the same rotation. Converting between them is a matter of scaling by the ratio between these full-circle values, using radians (the natural mathematical unit tied directly to a circle's radius) as the common base.
Example Calculation
An angle of 45° converts to 45 × (π/180) ≈ 0.785 radians, or 45 × (10/9) = 50 gradians.
How to Use
- Enter an angle value.
- Select the unit (degree, radian, or gradian).
- View the converted angle across all supported units instantly.
Benefits
- Converts all three angle units at once instead of one pair at a time.
- Avoids manual π-based calculation errors.
- Useful across math, physics, engineering, and surveying contexts.
Use Cases
- Converting calculator or textbook answers between degrees and radians.
- Working with gradians in surveying or European engineering contexts.
- Checking trigonometric function inputs that expect radians.
What Your Result Means
All three values represent the exact same angle, just expressed on different scales — 360°, 2π radians, and 400 gradians all correspond to one full rotation.
Tips
- Remember that most programming languages and scientific calculators expect radians for trig functions, not degrees.
- Use exact fractions of π (like π/2, π/4) when precision matters more than decimal approximations.
- Gradians are rarely used outside surveying and certain European contexts — degrees or radians cover most needs.
Common Mistakes
- Feeding a degree value into a function that expects radians (a common bug in trigonometric code).
- Confusing gradians (400 per circle) with degrees (360 per circle) when reading old surveying data.
- Rounding intermediate radian values too early, compounding error in further calculations.
FAQs
What is a radian?
A radian is the standard unit of angle measurement in mathematics, defined as the angle subtended when the arc length equals the radius of a circle.
Why are radians used instead of degrees in math?
Radians simplify calculus and trigonometric formulas, since they relate directly to the radius and arc length of a circle without an arbitrary scaling factor like 360.
What is a gradian?
A gradian (or gon) divides a full circle into 400 units instead of 360, making a right angle exactly 100 gradians — occasionally used in surveying and some European countries.
How many radians are in a full circle?
A full circle is exactly 2π radians, approximately 6.2832 radians.
Do programming languages use degrees or radians?
Most programming languages (JavaScript, Python, etc.) use radians for trigonometric functions like sin() and cos(), so degree values usually need converting first.
What is a right angle in each unit?
A right angle is 90 degrees, π/2 radians (≈1.5708), or 100 gradians.
Can angle values be negative?
Yes, negative angles represent rotation in the opposite (clockwise) direction and convert the same way as positive angles.
Conversions are exact mathematical relationships based on π.
Last updated: July 25, 2026