Angle Conversion Guide
Degree, Arc Minute & Arc Second
A degree (°) is 1/360 of a full rotation. A degree is divided into 60 arc minutes ('), and each arc minute is divided into 60 arc seconds ("). These divisions are widely used in astronomy, geography, and navigation.
Radian & Milliradian
A radian (rad) is the standard unit of angular measure used in mathematics and physics. A full circle is 2π radians. The milliradian (mrad, 1/1000 rad) is highly useful in optics and ballistics, where 1 mrad translates to 1 meter offset at a distance of 1000 meters.
Gradian, NATO Mil & Turn
A gradian (grad or gon) is 1/400 of a full turn, originally designed to fit decimal systems. NATO mils represent 1/6400 of a full turn, used in tactical sighting. A turn represents one complete 360-degree rotation.
Key Unit Relationships & Formulas
| Unit Relationship | Formula | Value in Degrees |
|---|---|---|
| 1 Degree (°) | Base Unit | 1° |
| 1 Radian (rad) | 180° / π | ≈ 57.29577951° |
| 1 Arc Minute (') | 1° / 60 | ≈ 0.01666667° |
| 1 Arc Second (") | 1° / 3600 | ≈ 0.00027778° |
| 1 Gradian (grad) | 360° / 400 | 0.9° |
| 1 Milliradian (mrad) | 180° / (1000 * π) | ≈ 0.05729578° |
| 1 NATO Mil (mil) | 360° / 6400 | 0.05625° |
| 1 Turn (turn) | 360° | 360° |
Frequently Asked Questions
What is the difference between a radian and a degree?
A degree is an arbitrary division (1/360 of a circle) dating back to ancient Babylon. A radian is a natural, dimensionless mathematical unit defined by the geometry of a circle (arc length divided by radius). 180° is exactly π radians.
How does the Degree-Minute-Second (DMS) representation handle negative angles?
The sign of the angle is carried entirely by the Degree field. For example, an angle of -10.5° is represented as -10 degrees, 30 minutes, and 0 seconds. If the angle is between 0° and -1° (e.g., -0.25°), it shows as -0 degrees, 15 minutes, and 0 seconds.
Why does the trigonometric ratios section show "Undefined" for some angles?
Certain trigonometric functions are ratios that involve division by zero at specific angles. For example, tan(θ) = sin(θ)/cos(θ), and at 90° (or 270°), cos(θ) is exactly 0. Since dividing by zero is mathematically undefined, these values are output as "Undefined".