Angle Unit Converter | Convert between Degrees, Radians, Percent and more

Angle Unit Basics

An angle is a physical quantity used to measure the size of the angle between two rays. Common angle units include degrees (°), radians (rad), grads, etc. The degree is the most commonly used angle unit, with a complete circle equal to 360 degrees.

Relationship Between Degrees and Radians

• The radian is the standard angle unit in the International System of Units. One radian equals the central angle subtended by an arc whose length equals the radius. A full circle is 2π radians, so: 1 radian = 180°/π ≈ 57.2958°
• 1 degree = π/180 radians ≈ 0.01745 radians.

Degree-Minute-Second System

Degrees-minutes-seconds is a precise representation of angles, commonly used in geographic coordinates, astronomy, and surveying. 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds. For example: 45°30'15" represents 45 degrees, 30 minutes, 15 seconds. This system can be precise to the arcsecond level.

Grad Unit

The grad (also called gon) is another angle unit. A complete circle equals 400 grads. 1 grad = 0.9 degrees = π/200 radians. The grad unit is relatively common in surveying and engineering applications in some European countries.

Angle Unit Conversions

• Common angle unit conversions: 360 degrees = 2π radians = 400 grads = 1 turn
• 90 degrees = π/2 radians = 100 grads = 1/4 turn
• 180 degrees = π radians = 200 grads = 1/2 turn
• 1 milliradian = 0.001 radians ≈ 0.0573 degrees.

Angles in Different Fields

• Geometry and trigonometry mainly use degrees and radians
• Geography and navigation commonly use degree-minute-second format for latitude and longitude
• Engineering selects appropriate units based on precision requirements
• Physics calculations typically use radians
• Computer graphics commonly use degrees for rotation angles
• Astronomy uses degrees-minutes-seconds for celestial positions.

Angle Measurement Precision

• Different applications require different angle precision: General measurements use degrees with ~0.1° precision
• Engineering surveys use degree-minute-second with 1" precision
• High-precision measurements use radians with micro-radian precision
• GPS positioning systems achieve angle precision within 0.1"
• Astronomical observations can achieve milli-arcsecond precision.