Angle Unit Converter: degrees, radians and gradians

This calculator converts angles between the three most commonly used units: degrees, radians and gradians (centesimal degrees), essential in mathematics, physics and engineering.

Conversion formulas

The relationships between units are:

• Degrees to radians: rad = degrees × π / 180
• Radians to degrees: degrees = rad × 180 / π
• Degrees to gradians: grad = degrees × 10 / 9
• Gradians to degrees: degrees = grad × 9 / 10

A full circle = 360° = 2π rad ≈ 6.283 rad = 400 grad.

Example 1: degrees to radians

Problem: Convert 45° to radians.

1. Calculation:
   • rad = 45 × π / 180 = π / 4 ≈ 0.7854 rad.

Answer: 45° = π/4 ≈ 0.7854 rad.

Example 2: radians to degrees

Problem: Convert 1.5 rad to degrees.

1. Calculation:
   • degrees = 1.5 × 180 / π ≈ 85.94°.

Answer: 1.5 rad ≈ 85.94°.

Common uses of angle conversion

• Working with trigonometric functions in calculators and software.
• Solving physics problems that require radians.
• Converting measurements in surveying.
• Programming graphics and animations in web development.
• Computing in mechanical engineering and robotics.
• Studying mathematics and trigonometry.

Common mistakes with angles

• Using degrees when the trig function expects radians.
• Confusing radians with gradians.
• Not checking calculator mode (DEG vs RAD).
• Rounding π too early in calculations.

Pro tip

In calculus and mathematical analysis, radians are the natural unit. The derivatives of sin(x) and cos(x) are only correct when x is in radians. Memorize key angles: 0°, 30°, 45°, 60°, 90° and their radian equivalents.