My rotary encoder outputs positions in arcseconds but my motion controller expects turns. How do I convert?

Divide arcseconds by 1,296,000. For a position of 324,000 arcseconds: 324,000 ÷ 1,296,000 = 0.25 turns (90°). Most motion controllers then multiply turns by steps-per-revolution to get step counts.

An astronomical catalog gives a polar axis misalignment of 2,400 arcseconds. What fraction of a turn is that?

2,400 ÷ 1,296,000 ≈ 0.00185 turns (= 0.667°). That's about 1/540th of a full rotation — a small but meaningful misalignment for long-exposure astrophotography where tracking drift matters.

Why express tiny angles as turns instead of keeping them in degrees or arcseconds?

Turns are natural for rotary systems: 1 turn = 1 full revolution regardless of unit system. When commanding stepper motors or servo drives, turns multiply directly by steps/revolution or encoder counts/revolution without additional degree-to-radian conversions.

Can I estimate turns from arcseconds quickly?

Divide by 1.3 million for a rough estimate. For 650,000 arcseconds: 650,000 ÷ 1,300,000 ≈ 0.5 turns. The exact divisor is 1,296,000, so this gives about 0.3% error — fine for quick checks.

Arcseconds to Turns Converter

Enter a value to instantly convert between angle units.

From

Precision4 dp

1 Arcsecond = 7.7160e-7 Turn

Key Formulas

Degree → Radian

rad = ° × 0.0174533

Radian → Degree

° = rad × 57.2958

Degree → Gradian

grad = ° × 1.11111

Gradian → Degree

° = grad × 0.9

Formula

turns = arcseconds ÷ 1296000

CNC rotary tables and precision indexing heads record angular positions in arcseconds for maximum resolution, but motion controllers and G-code programs specify rotation as fractional turns or decimal revolutions. A rotary encoder with 0.5 arcsecond resolution produces position data that must be converted to turns before a controller can command the axis to the correct position. The factor 1,296,000 comes from 360° × 60′ × 60″.

Source: ISO 80000-3:2019 (Quantities and units — Space and time)

Last reviewed: 2026-06-11 · see our methodology

Frequently Asked Questions

Real-World Examples

1,296,000 arcseconds — exactly 1 full turn (360°). The anchor conversion establishing the complete arcsecond-to-turn relationship.

1296000 ″ = 1 tr

648,000 arcseconds (180°) — exactly half a turn, the position a rotary stage must reach to flip a workpiece to the opposite face for machining.

648000 ″ = 0.5 tr

3,600 arcseconds (1°) — 0.00278 turns, one degree expressed as a fraction of a full rotation for a servo-driven antenna positioning system.

3600 ″ = 0.002778 tr

1,800 arcseconds (0.5°) — 0.00139 turns, a fine angular correction entered into a telescope pointing model as a fractional revolution.

1800 ″ = 0.001389 tr

degrees to radians radians to degrees degrees to gradians gradians to degrees

Arcseconds to Turns Converter

Key Formulas

Formula

Frequently Asked Questions

Real-World Examples

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