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Angle Converter: Degrees, Radians, and Gradians Explained

By David Brown · January 2026 · 3 min read

Angles can be measured in degrees, radians, or gradians. Most people only encounter degrees in daily life, but radians are the standard in calculus and physics, and gradians appear in surveying and some European traditions.

Degrees

The familiar system: a full circle is 360°. Right angle: 90°. Straight line: 180°.

Why 360? Historically attributed to the approximately 360 days in a year and the divisibility of 360 by many small integers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12...).

Radians

In radians, a full circle is 2π ≈ 6.2832 radians. A right angle is π/2 ≈ 1.5708 radians.

Why radians? Because they make the math of circular motion, trigonometry, and calculus cleaner. The derivative of sin(x) is cos(x) — but only when x is in radians. In degrees, it's (π/180)cos(x). Radians eliminate that constant.

Key conversions:

0° = 0 rad
90° = π/2 rad ≈ 1.5708
180° = π rad ≈ 3.1416
270° = 3π/2 rad ≈ 4.7124
360° = 2π rad ≈ 6.2832

Formula: radians = degrees × (π/180)

Gradians (Gon)

A full circle is 400 gradians. A right angle is exactly 100 gradians. Used in surveying, civil engineering, and military applications in some countries because the decimal relationship simplifies certain calculations.

The advantage: 45° = 50 gradians, 90° = 100 gradians — cleaner numbers for slope calculations.

When You'll Actually Use Radians

If you're writing trigonometric functions in code (Python, JavaScript, etc.), the math functions take radians, not degrees. math.sin(90) gives you the wrong answer; math.sin(math.pi/2) gives you 1.0.

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