Angle Converter

About the Angle Converter

The Angle Converter converts between degrees, radians, gradians (gons), turns (revolutions), and arcminutes/arcseconds. Enter an angle in any unit and review the equivalents with precise mathematical accuracy.

Angles are measured in different units depending on the field. Degrees (0–360) are used in everyday geometry, navigation, and engineering. Radians (0–2π) are the mathematical standard used in calculus, physics, and programming. Gradians (0–400) are used in surveying and some European countries. Turns (0–1) represent full revolutions.

The core relationship is: radians = degrees × π/180. Our converter applies this precisely using JavaScript's Math.PI constant for maximum accuracy. It's essential for students, engineers, programmers, and scientists working across different angle conventions.

When This Page Helps

Math libraries use radians, textbooks use degrees, surveyors use gradians. This converter bridges all systems with exact mathematical precision for cross-unit reference.

How to Use the Inputs

1. Select the angle unit you are converting from.
2. Enter the angle value.
3. View the equivalent values in all other angle units.
4. Use radians for programming and calculus functions.
5. Use degrees for geometry, navigation, and practical measurements.

Example Calculation

Tips & Best Practices

• Key values: 0°=0 rad, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 360°=2π.
• Most programming languages (JavaScript, Python, C) use radians for trig functions like sin(), cos(), tan().
• To convert in code: radians = degrees * Math.PI / 180.
• Gradians divide a right angle into exactly 100, making surveying calculations cleaner.
• Negative angles and angles > 360° are valid and represent direction or multiple rotations.

Angle Units in Context

Degrees originated in ancient Babylon (base-60 system). Radians were formalized by mathematicians for their elegant properties in calculus. Gradians were introduced during the French Revolution as part of the metric push. Each unit serves its field optimally.

Radians in Programming

JavaScript's Math.sin(), Math.cos(), Math.atan2() all expect radians. Python's math module is the same. Always convert degrees to radians before calling trig functions: Math.sin(degrees * Math.PI / 180). Some libraries provide toRadians() and toDegrees() helpers.

Common Angle Reference Points

0° = start, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2 (right angle), 180° = π (straight line), 270° = 3π/2, 360° = 2π (full circle). Memorizing these is essential for trigonometry and physics.

Frequently Asked Questions

• How do I convert degrees to radians?

  Multiply the degree value by π/180 (approximately 0.01745). For example, 45° × π/180 = π/4 ≈ 0.7854 radians. To go from radians to degrees, multiply by 180/π.

• Why do mathematicians use radians instead of degrees?

  Radians simplify calculus formulas. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, the derivative includes a π/180 factor. Radians make formulas cleaner because they measure arc length on the unit circle directly.

• What is a gradian?

  A gradian (also called gon or grad) divides a right angle into 100 equal parts. A full circle = 400 gradians. This makes percentage calculations easy: a 1% slope = 1 gradian. Gradians are used in surveying, particularly in Europe.

• What is a turn?

  One turn (or revolution) equals 360 degrees = 2π radians = 400 gradians. It's the most intuitive unit for full rotations. 0.5 turns = 180° (half turn), 0.25 turns = 90° (quarter turn).

• How many radians in a full circle?

  A full circle contains exactly 2π radians (≈ 6.2832). A half circle = π radians. A quarter circle (right angle) = π/2 radians. One radian ≈ 57.296 degrees.

• What are arcminutes and arcseconds?

  One degree = 60 arcminutes (60'), and one arcminute = 60 arcseconds (60"). They provide finer angular resolution. Used in astronomy (star positions), navigation (latitude/longitude), and optics.