Introduction
Angles are measured in degrees, radians, or gradians, with each unit preferred by a different profession. Degrees dominate everyday use (360 per circle, inherited from Babylonian astronomy). Radians dominate math and physics (2π per circle, the natural unit for calculus). Gradians (400 per circle) show up in surveying and artillery. Navigation and astronomy subdivide degrees into arc minutes and arc seconds.
Why angle units exist and how they diverged
The 360-degree circle comes from Babylonian astronomy, where the year was close to 360 days. A day was thus 1 degree of Earth's orbit. The factor 360 has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180), which made mental math easy without a calculator. The radian, adopted by 18th-century mathematicians, is defined so that the arc length of a unit circle equals the angle: 1 radian = 1 unit of arc, meaning d(sin x)/dx = cos x only in radians. It makes calculus clean.
Gradians were a French Revolution innovation (1790s), along with decimal time and decimal angles. Decimal time failed; gradians survived in surveying, where a right angle of 100 gradians divides cleanly.
How to convert angle
Degrees to radians: multiply by π/180 ≈ 0.01745. Radians to degrees: multiply by 180/π ≈ 57.296. Degrees to gradians: multiply by 10/9. 1 arc minute = 1/60 degree; 1 arc second = 1/3600 degree. Common angles in radians: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.
Units supported by this angle calculator
- Degrees
- Radians
- Gradians
- Turns
- Arc minutes
- Arc seconds
Common angle conversion mistakes
- Calculator mode. Scientific calculators and spreadsheets default to radians. A 'sin(90)' in radians gives 0.894, not 1 — because 90 rad is ~14 full rotations plus an offset. Always check mode.
- Arc minutes vs decimal degrees. 40.5 degrees is 40 degrees 30 minutes, not 40 degrees 50 minutes. The decimal is a fraction of the full degree, not of 60.
- True north vs magnetic north. Compasses read magnetic north; maps use true north. Magnetic declination varies by location (about 0-20° in most of North America). GPS gives both; compass users need to correct.
- Bearings vs heading. Bearing is measured from north, clockwise. Heading is the direction something is pointing. 'North 90 east' (bearing) is the same as 'heading east.' Aviation uses compass headings (360° instead of 0° for north).
- Slope grade vs angle. A road with a 10% grade has an angle of 5.7°, not 10°. Grade = rise/run × 100%; angle = arctan(rise/run).
Real-world angle examples
- Right angle: 90° = π/2 rad = 100 grad.
- Straight line: 180° = π rad = 200 grad.
- Full circle: 360° = 2π rad = 400 grad.
- Internal angle of equilateral triangle: 60° = π/3 rad.
- Compass directions: N = 0°, E = 90°, S = 180°, W = 270°.
- One arc minute of Earth latitude ≈ 1 nautical mile (1,852 m).
- Moon's angular diameter from Earth: ~0.52° (about 31 arc minutes).
- Sun's angular diameter from Earth: ~0.53° (coincidentally close to the moon, which is why solar eclipses work).
Tips for accurate angle conversion
- For CAD and machining, prefer decimal degrees to minutes/seconds — cleaner arithmetic.
- In spreadsheets, use
=RADIANS(degrees)and=DEGREES(radians)to convert for trig functions. - For GPS coordinates, always specify decimal degrees or DMS (degrees/minutes/seconds). Mixing confuses maps.
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