Angle Conversion Calculator
Calculator and formulas for angle units of measurement (Degree, Radian, Gradian, Mil)
Angle Unit Calculator
Angle Converter
Select the unit of measurement that you know and enter its value. The calculator supports degrees, radians, gradians, and mil (military).
Angle Units - Overview
Degree System
Full circle = 360° = 2π rad = 400 gon
Degree
360°
Full circle
DecimalRadian
2π rad
Full circle
≈ 6.283Important Conversions
- 1° = 60' = 3,600"
- π rad = 180° = 200 gon
- 1 rad ≈ 57.296° ≈ 63.662 gon
- 1 gon = 0.9° = 54'
- 1 mil ≈ 0.05625° (NATO)
Degree System
- 1° (degree) = base unit
- 1' (arcminute) = 1/60° ≈ 0.01667°
- 1" (arcsecond) = 1/3600° ≈ 0.000278°
- Full circle = 360°
- Right angle = 90°
Radian (Arc measure)
- 1 rad = 180°/π ≈ 57.296°
- π rad = 180° (semicircle)
- 2π rad = 360° (full circle)
- π/2 rad = 90° (right angle)
- Preferred in mathematics and physics
Gradian (Gon/New degree)
- 1 gon = 0.9° = 54'
- Full circle = 400 gon
- Right angle = 100 gon
- Decimal division: 1 gon = 100 cgon
- Used in surveying
Mil (Military)
- NATO: 1 mil ≈ 0.05625° (6400 mil = 360°)
- USSR: 1 mil = 0.06° (6000 mil = 360°)
- Sweden: 1 streck = 0.05625°
- Used in artillery and navigation
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Formulas of Angle Conversion
Basic Conversions
Degree, Arcminute, Arcsecond
\(\displaystyle 1° = 60' = 3600''\)
From Radian to Degree and Gradian
\(\displaystyle degree = radian \cdot \frac{180}{\pi}\)
\(\displaystyle gradian = radian \cdot \frac{200}{\pi}\)
From Degree to Radian and Gradian
\(\displaystyle radian = degree \cdot \frac{\pi}{180}\)
\(\displaystyle gradian = degree \cdot \frac{100}{90}\)
Extended Conversions
From Gradian to Radian
\(\displaystyle radian = gradian \cdot \frac{\pi}{200}\)
Mil Conversions
\(\displaystyle mil = degree \cdot \frac{3200}{180}\)
\(\displaystyle mil = gradian \cdot 16\)
Note
Radian is the SI unit for angles. It is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. 2π radians equal 360 degrees (one full circle).
Comparison Table of Important Angles
| Description | Degree (°) | Radian (rad) | Gradian (gon) | Mil (NATO) |
|---|---|---|---|---|
| Zero angle | 0° | 0 | 0 gon | 0 mil |
| Right angle | 90° | π/2 ≈ 1.571 | 100 gon | 1,600 mil |
| Straight angle | 180° | π ≈ 3.142 | 200 gon | 3,200 mil |
| Full angle | 360° | 2π ≈ 6.283 | 400 gon | 6,400 mil |
| 45° angle | 45° | π/4 ≈ 0.785 | 50 gon | 800 mil |
| 60° angle | 60° | π/3 ≈ 1.047 | 66.67 gon | 1,067 mil |
| 30° angle | 30° | π/6 ≈ 0.524 | 33.33 gon | 533 mil |
Practical Examples
Common Angles:
- Clock: 12:15 o'clock → 90° between hands
- Traffic sign: yield → 60° interior angle
- Roof pitch: 45° = 100% slope
- Sun angle: 23.5° Earth's axial tilt
Technical Applications:
- Geodesy: Gradian (gon) preferred
- Mathematics/Physics: Radian (rad) standard
- Military/Artillery: Mil (NATO, USSR)
- Astronomy: Degree, minutes, seconds (HMS)
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