Convert Degrees (°) to Milliradians (mrad)

What Is a Degree?

The degree (°) is a non-SI unit of plane angle defined as exactly 1/360 of a full rotation. It is accepted for use with the SI due to its importance in navigation, surveying, engineering, and astronomy.

The degree is the most common angular unit for everyday measurement — compass headings, protractor readings, GPS coordinates, and construction all use degrees. For applications requiring sub-degree precision, milliradians offer a metric-based alternative.

Key equivalences: 1° = π/180 rad ≈ 17.453 mrad = 60′ = 3,600″.

What Is a Milliradian?

The milliradian (mrad) is one-thousandth of a radian. Since 1 rad ≈ 57.2958°, there are approximately 17.453 milliradians per degree. A full circle contains about 6,283 mrad.

The milliradian's key practical property: at any distance, 1 mrad subtends a length equal to 1/1,000 of that distance. At 100 m, 1 mrad = 10 cm; at 1 km, 1 mrad = 1 m. This linear relationship makes milliradians the preferred unit in precision shooting, optical engineering, and laser alignment.

Key equivalences: 1 mrad = 0.001 rad ≈ 0.05730°; 1,000 mrad = 1 rad.

Degrees to Milliradians Formula

The conversion factor derives from the radian definition:

milliradians = degrees × 1000π/180 ≈ degrees × 17.4533

Since 1° = π/180 rad and 1 mrad = 0.001 rad, multiplying by 1000π/180 gives the exact result.

Worked Example

Convert a 1 MOA (1/60°) scope adjustment to milliradians:

1. Convert: 1/60° = 0.01667°
2. Multiply: 0.01667 × 17.453 = 0.2909 mrad

So 1 MOA ≈ 0.291 mrad, which means at 100 m, 1 MOA subtends about 2.91 cm — compared to 10 cm for 1 mrad.

Quick Estimation

For mental math, multiply degrees by 17.5 — this overshoots by only 0.27%. Example: 5° × 17.5 = 87.5 mrad (exact: 87.27 mrad). For the reverse, divide mrad by 17.5 to get approximate degrees.

Common Conversions at a Glance

Which Is Bigger: Degree or Milliradian?

A degree is about 17.5 times larger than a milliradian. A degree covers a visible slice of a circle; a milliradian is a fine subdivision used for precision work. At 1 km, 1 degree spans about 17.5 meters, while 1 mrad spans exactly 1 meter.

When You Need This Conversion

• Precision shooting: Mil-dot scopes use milliradians for reticle subtension and turret adjustments. Comparing mrad-based scopes with MOA-based specifications requires degree-to-mrad conversion.
• Optical engineering: Laser beam divergence and telescope pointing accuracy are specified in mrad. A typical laser pointer diverges ~1 mrad, creating a 1 m spot at 1 km.
• Robotics and motion control: Servo motors and gimbal systems specify pointing accuracy in mrad. Satellite tracking typically achieves 0.1–1 mrad.
• Automotive LIDAR: Self-driving sensors specify angular resolution in mrad. Typical automotive LIDAR: 1–5 mrad resolution, determining object discrimination at distance.