Angle Units Explained: Degrees, Radians and Gradians

Angles are usually measured in degrees in everyday life, but radians are the unit math and programming actually run on behind the scenes, and gradians show up in a handful of surveying and engineering contexts. Knowing all three, and why radians exist at all, avoids a genuinely common bug.

Common Angle Units

• Degree (°): the everyday unit, with a full circle equal to 360°.
• Radian (rad): the mathematical unit, with a full circle equal to 2π radians.
• Gradian (grad): a less common unit, with a full circle equal to 400 gradians, used in some surveying and engineering work.

Why Radians Exist

A radian is defined so that an angle of 1 radian traces an arc length exactly equal to the circle’s radius. That direct link between angle and arc length is what makes radians the natural unit for calculus and trigonometry, formulas involving angles come out simpler in radians than in degrees, which is why programming languages and math libraries default to them.

The Conversion Formulas

Worked Example

Common Mistakes

• Passing degrees into a function that expects radians. Most programming languages’ built-in trig functions (sin, cos, tan) expect radians, not degrees; passing degrees directly produces silently wrong results rather than an error.
• Assuming gradians are rare enough to ignore entirely. They still appear in some surveying, artillery, and older calculator “GRAD” modes, so it’s worth recognizing the unit even if you rarely use it.
• Mixing up a full turn’s value across units. 360° = 2π radians = 400 gradians; forgetting which full-circle value belongs to which unit is an easy slip.

Use the Angle Converter

The Angle Converter converts between degrees, radians, gradians, and more instantly.