Free Angle Converter – Degrees to Radians, DMS & 7 Units

180 degrees = π radians (≈ 3.14159 radians). So 1 radian ≈ 57.2958°. Full circle: 360° = 2π rad ≈ 6.283 rad. Conversion: multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees.

Radians are dimensionless (pure ratio), making calculus and trigonometry work naturally. Derivatives of sin/cos require radian input. All scientific calculators, programming languages (Python, JavaScript, C++), and physics engines use radians by default. Degrees are human-friendly but mathematically arbitrary.

Subdivisions of degrees: 1° = 60 arcminutes (′) = 3600 arcseconds (″). Used in astronomy, surveying, and GPS for high precision. Example: Star position 5″ = 1/720 of a degree. On Earth: 1 arcsecond ≈ 30 meters at equator. Essential for telescope accuracy.

Gradian (gon) divides full circle into 400 parts (instead of 360 degrees). 1° = 1.111 gradians; 90° = 100 gradians. Used in European surveying, particularly France and Switzerland. Less common globally than degrees or radians.

One complete rotation = 1 revolution = 1 turn = 360° = 2π radians ≈ 6.283 radians. Used for multiple rotations or rotational measurements. Example: 2.5 revolutions = 900° = 5π radians = 2.5 full spins.

45° × (π/180) = π/4 ≈ 0.7854 radians. Formula: radians = degrees × π/180. Example: 90° = π/2 ≈ 1.5708 rad; 180° = π ≈ 3.1416 rad; 360° = 2π ≈ 6.2832 rad.

1000 milliradians = 1 radian. 1 mrad ≈ 0.0573° ≈ 3.44 arcminutes. Military advantage: 1 mrad = 1 meter at 1000 meters distance (easy mental math for rifle scope adjustments). Used in ballistics, artillery, laser targeting systems.

gradians = degrees × (10/9). Example: 90° = 100 gradians; 180° = 200 gradians; 360° = 400 gradians. Reverse: degrees = gradians × (9/10). Used primarily in European surveying (France, Switzerland).

Decimal degrees = degrees + (minutes/60) + (seconds/3600). Example: 45°30′15″ = 45 + (30/60) + (15/3600) = 45.504167°. Used in GPS and surveying. Reverse: extract degrees (integer part), then minutes and seconds from decimal remainder.

Arc length = radius × angle (in radians). Formula: s = r × θ. Example: Circle radius 5 meters, arc angle 2 radians: arc length = 5 × 2 = 10 meters. This formula only works with radians (not degrees). Key for circular motion physics.

All major languages (Python, JavaScript, C++, Java) use radians because: (1) Mathematics foundation uses radians; (2) Calculus derivatives require radians; (3) Physics simulations standardize on radians; (4) Converting at runtime wastes CPU. Always convert degrees to radians BEFORE calling sin/cos/tan.

1 degree ≈ 111 kilometers. 1 arcminute ≈ 1.85 kilometers. 1 arcsecond ≈ 30 meters. Modern GPS: ±5 arcseconds horizontal accuracy (±150 meters). Surveying: ±0.1 arcsecond achievable with RTK-GPS (±3 meters). Astronomy: telescopes measure arcseconds for target precision.

Use haversine formula to calculate bearing: θ = atan2(sin(Δλ)cos(φ2), cos(φ1)sin(φ2) - sin(φ1)cos(φ2)cos(Δλ)). Δλ = longitude difference; φ = latitudes. Result in radians; convert to degrees (multiply by 180/π). This converter handles angle unit conversions; haversine requires coordinate calculator.

A radian is the angle made when arc length equals the radius. Draw circle, arc 1 = radius 1, the angle between them = 1 radian. Full circle has 2π radians (≈ 6.283) because circumference = 2πr. Intuitive once visualized; mathematically perfect for calculus.

Hubble Space Telescope: 0.05 arcsecond accuracy (1/72,000 Moon's width). Ground telescopes: 0.1–1 arcsecond depending on conditions. Astrometry catalogs (Gaia): 0.001 arcsecond precision for star positions. Arcsecond differences = different targets in space. Essential for astronomical observation success.

Semicircle arc length = half circumference = πr. Angle for semicircle = arc length / radius = πr / r = π radians. Since semicircle = 180°, therefore 180° = π radians. Full circle: 360° = 2π radians (full circumference 2πr ÷ r = 2π).