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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/180rad=10/9grad1° = π/180 rad = 10/9 grad

How It Works

This calculator converts between degrees, radians, gradians, arcminutes, arcseconds, and turns for engineers, surveyors, and scientists working with angular measurements. Per SI Brochure (BIPM) and NIST SP 811, the radian is the SI unit for plane angle: 1 rad = 180/pi degrees = 57.29578 degrees exactly. One full revolution = 360 degrees = 2*pi radians = 400 gradians = 21,600 arcminutes = 1,296,000 arcseconds = 1 turn. Precision matters: GPS positioning uses arcseconds where 1 arcsecond of latitude = 30.87 meters on Earth's surface (WGS84 ellipsoid). Antenna beamwidth is specified in degrees (typical 3 dB beamwidth: 5-60 degrees), while stepper motors use steps per revolution (200 steps = 1.8 degrees/step per NEMA standard).

Worked Example

Problem

A phased array antenna requires beam steering of 45 degrees with 0.1 degree resolution. Calculate in all angular units and determine required phase shifter bits.

Solution
  1. Convert 45 degrees: 45 × pi/180 = 0.7854 rad = 50 grad = 2700 arcmin = 162,000 arcsec = 0.125 turns
  2. Resolution 0.1 degrees: 0.1 × pi/180 = 1.745 mrad = 6 arcmin = 360 arcsec
  3. Steps needed: 45/0.1 = 450 discrete positions
  4. Bits required: ceiling(log2(450)) = 9 bits for 512 positions
  5. Phase resolution per IEEE 802.11ad: lambda/2 spacing, 45 degrees = pi/4 rad phase difference
  6. 9-bit phase shifter provides 360/512 = 0.703 degree resolution (better than 0.1 degree requirement)

Practical Tips

  • ✓Per IEEE/ISO standards: store angles in radians internally, convert to degrees only for display. This eliminates conversion overhead in trigonometric calculations and matches library expectations (sin, cos, atan2)
  • ✓Antenna specifications per IEEE 145-2013: 3 dB beamwidth in degrees, pointing accuracy in arcminutes or arcseconds. A 1 degree pointing error at 36,000 km (GEO satellite) = 628 km position error
  • ✓Stepper motor resolution per NEMA: 200 full steps/rev = 1.8 deg/step, with 256x microstepping = 0.007 deg/microstep. Convert to radians for trajectory planning: 1.8 deg = 31.42 mrad

Common Mistakes

  • ✗Using degrees in trigonometric functions expecting radians - Math.sin(90) in most languages gives 0.894 (90 radians), not 1.0 (90 degrees). Convert first: Math.sin(90 × pi/180) = 1.0
  • ✗Confusing gradians (400 per circle) with degrees (360 per circle) - causes 11.1% error. 100 grad = 90 degrees, not 100 degrees. Gradians appear in some European surveying equipment
  • ✗Accumulating rounding errors in repeated angle calculations - after 360 additions of 1 degree, floating-point error can reach 10^-14 radians. Wrap angles modulo 2*pi periodically

Frequently Asked Questions

Radians make formulas cleaner per SI Brochure: arc length s = r × theta (exact for theta in radians), angular velocity omega = 2*pi*f rad/s, Taylor series sin(x) = x - x^3/6 + ... (valid for x in radians). The radian is dimensionless (m/m), eliminating unit conversion factors in derivatives: d/dx sin(x) = cos(x) only when x is in radians.
A gradian (gon) = 1/400 circle, so 90 degrees = 100 grad (a right angle is exactly 100). Per ISO 31-1, gradians simplify grade calculations: a 1% slope = 1 grad. Used in European surveying (theodolites, total stations) and some calculators. Rarely seen in electronics or RF engineering.
1 arcminute = 1/60 degree = 0.2909 mrad = 0.000291 rad. This is approximately the angular resolution limit of human vision (20/20 = 1 arcmin). Telescope pointing: 1 arcmin at Moon distance (384,400 km) = 112 km. Antenna pointing per ITU-R: a 1 arcmin error for a 30 dBi dish is acceptable.
Standard NEMA stepper: 200 steps/rev = 1.8 deg/step = 31.42 mrad/step. With 16x microstepping: 3200 microsteps/rev = 0.1125 deg/microstep = 1.96 mrad/microstep. For precision: 256x microstepping gives 0.007 deg = 25 arcsec resolution, suitable for telescope drives and CNC machines.

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