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Angle Unit Converter

Convert angles between degrees, radians, gradians, arcminutes, arcseconds, and turns for motor, antenna, and RF applications.

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Formula

1° = π/180 rad = 10/9 grad

How It Works

Angles can be expressed in degrees (°), radians (rad), gradians (grad), arcminutes ('), arcseconds ("), and turns (revolutions). Radians are the SI-preferred unit for mathematical and signal processing calculations (1 revolution = 2π rad). Degrees are universal in engineering practice. Gradians (400 per circle) are used in surveying. Arcminutes and arcseconds appear in precision optical and astronomical applications.

Worked Example

A quarter-turn servo rotation of 90°: 90° = π/2 rad ≈ 1.5708 rad = 100 grad = 5400 arcminutes = 324,000 arcseconds = 0.25 turns. A small beam steering angle of 0.1°: 0.1° = 0.001745 rad = 0.111 grad = 6 arcminutes = 360 arcseconds.

Practical Tips

  • ✓In embedded firmware, convert once at input/output boundaries: store and compute in radians internally, display in degrees for users.
  • ✓Antenna beamwidth is quoted in degrees (e.g., 3 dB beamwidth of 12°); servo and stepper positions are often in encoder counts that map to degrees.
  • ✓Arcminutes and arcseconds matter for precision optics and GPS — 1 arcsecond of latitude ≈ 30.8 m on Earth's surface.

Common Mistakes

  • ✗Using degrees instead of radians in trigonometric calculations in code — most math libraries (sin, cos, atan2) expect radians.
  • ✗Confusing gradians with degrees — 100 grad = 90°, not 100°. Gradians are only common in European surveying equipment.
  • ✗Accumulating floating-point errors in angle calculations; after many additions, wrap the angle modulo 360° (or 2π) to prevent drift.

Frequently Asked Questions

Radians make many formulas cleaner: arc length = r × θ (in rad), angular velocity ω = 2πf (in rad/s), and Fourier transforms naturally use radians. Degrees require conversion factors that clutter equations.
A gradian (also called gon) is 1/400 of a full circle, so a right angle = 100 grad. Gradians are used in surveying instruments, particularly in European civil engineering, because they simplify grade calculations.
One arcminute (1/60 of a degree) corresponds to about 0.00029 rad. It is the angular resolution of the human eye and the precision standard for many optical targeting and antenna pointing systems.
A common 200-step stepper motor moves 1.8° per full step (360°/200 = 1.8°). With 16× microstepping it achieves 0.1125° per microstep. These are directly entered in degrees for motion control calculations.

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