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Convert between all angle units: degrees, radians, gradians, turns, milliradians, and DMS format.
Degrees to Radians:
rad = deg × (π/180)
Degrees to Gradians:
gon = deg × (400/360)
Degrees to Turns:
turns = deg / 360
Degrees to Milliradians:
mrad = deg × (π×1000/180)
| Degrees (°) | Radians (rad) | Gradians (gon) | Turns |
|---|---|---|---|
| 0° | 0.000000 | 0.00000 | 0.00000 |
| 30° | 0.523599 | 33.33333 | 0.08333 |
| 45° | 0.785398 | 50.00000 | 0.12500 |
| 60° | 1.047198 | 66.66667 | 0.16667 |
| 90° | 1.570796 | 100.00000 | 0.25000 |
| 120° | 2.094395 | 133.33333 | 0.33333 |
| 180° | 3.141593 | 200.00000 | 0.50000 |
| 270° | 4.712389 | 300.00000 | 0.75000 |
| 360° | 6.283185 | 400.00000 | 1.00000 |
Degrees are the most common unit for measuring angles, dividing a circle into 360 equal parts. This ancient system, dating back to Babylonian mathematics, is intuitive and widely used in everyday applications, navigation, construction, and education. The 360-degree system was likely chosen because 360 has many divisors, making it easy to divide circles into halves, thirds, quarters, and more.
Radians are the standard unit in mathematics and science. One radian equals the angle where the arc length equals the radius. A full circle is 2π radians (≈6.28). Radians simplify many mathematical formulas, especially in calculus and trigonometry. They are essential for derivatives, integrals, and any advanced mathematical work involving angles.
Gradians divide a circle into 400 parts, making a right angle exactly 100 gradians. This decimal-based system integrates well with metric measurements and is used in surveying and civil engineering, particularly in Europe. The gradian system was designed to simplify calculations by using base-10 arithmetic throughout, from gradians to centigrades to milligrades.
A turn (also called a revolution or rotation) represents one complete rotation around a circle. One turn equals 360 degrees or 2π radians. This unit is intuitive for describing rotations in machinery, computer graphics, and physics. It's particularly useful when dealing with multiple full rotations, such as in gears, motors, and orbital mechanics.
Milliradians (mrad or mil) are one-thousandth of a radian, extensively used in military applications, rifle scopes, and artillery. The key advantage is that at 1000 meters, 1 milliradian subtends approximately 1 meter, making range estimation and ballistic calculations much simpler in the field. NATO forces use milliradians as the standard angular unit for fire control systems.
Use degrees for everyday applications, navigation, and construction. Use radians for mathematics, physics, and programming. Use gradians for European surveying. Use milliradians for military and shooting applications. Use turns for describing rotations and revolutions.
A full circle equals 360 degrees = 2π radians = 400 gradians = 1 turn = 6283.185 milliradians. All these units measure the same thing (angles) but use different scales. The choice depends on your application and which system makes calculations most convenient.
Different units emerged to solve specific problems. Degrees are historically established and intuitive. Radians simplify mathematical formulas. Gradians work well with metric systems. Milliradians make military range calculations easier. Each field adopted the unit that best served its needs, and these conventions persist today.
Our converter uses JavaScript's built-in Math.PI constant (accurate to 15-17 significant digits) for all calculations. The displayed results are rounded to 10 decimal places for readability, which is more than sufficient for virtually all practical applications, including scientific and engineering work.