An angle measures rotation or turn between two lines. Think of opening a door or turning a wheel. Measured in degrees (°), radians (rad), or gradians. 360° = full circle = one complete rotation.

• Angle = amount of rotation
• Full circle = 360° = 2π rad
• Right angle = 90° = π/2 rad
• Straight line = 180° = π rad

Degrees: circle divided into 360 parts (historical). Radians: based on circle radius. 2π radians = 360°. Radians are 'natural' for math/physics. π rad = 180°, so 1 rad ≈ 57.3°.

• 360° = 2π rad (full circle)
• 180° = π rad (half circle)
• 90° = π/2 rad (right angle)
• 1 rad ≈ 57.2958° (conversion)

Gradian: 100 grad = 90° (metric angle). Arcminute/arcsecond: subdivisions of degree (astronomy). Mil: military navigation (6400 mils = circle). Each unit for specific application.

• Gradian: 400 grad = circle
• Arcminute: 1′ = 1/60°
• Arcsecond: 1″ = 1/3600°
• Mil (NATO): 6400 mil = circle

360° per circle (Babylonian origin - ~360 days/year). Subdivided: 1° = 60′ (arcminutes) = 3600″ (arcseconds). Universal for navigation, surveying, everyday use.

• 360° = full circle
• 1° = 60 arcminutes (′)
• 1′ = 60 arcseconds (″)
• Easy for humans, historical

Radian: arc length = radius. 2π rad = circle circumference/radius. Natural for calculus (sin, cos derivatives). Physics, engineering standard. π rad = 180°.

• 2π rad = 360° (exact)
• π rad = 180°
• 1 rad ≈ 57.2958°
• Natural for math/physics

Gradian: 400 grad = circle (metric angle). 100 grad = right angle. Mil: military navigation - NATO uses 6400 mils. USSR used 6000. Different standards exist.

• 400 grad = 360°
• 100 grad = 90° (right angle)
• Mil (NATO): 6400 per circle
• Mil (USSR): 6000 per circle

rad = deg × π/180. deg = rad × 180/π. grad = deg × 10/9. Always use radians in calculus! Trig functions need radians for derivatives.

• rad = deg × (π/180)
• deg = rad × (180/π)
• grad = deg × (10/9)
• Calculus requires radians

sin, cos, tan relate angles to ratios. Unit circle: radius=1, angle=θ. Point coordinates: (cos θ, sin θ). Essential for physics, engineering, graphics.

• sin θ = opposite/hypotenuse
• cos θ = adjacent/hypotenuse
• tan θ = opposite/adjacent
• Unit circle: (cos θ, sin θ)

Angles add/subtract normally. 45° + 45° = 90°. Full rotation: add/subtract 360° (or 2π). Modulo arithmetic for wrapping: 370° = 10°.

• θ₁ + θ₂ (normal addition)
• Wrap: θ mod 360°
• 370° ≡ 10° (mod 360°)
• Negative angles: -90° = 270°

Compass bearings: 0°=North, 90°=East, 180°=South, 270°=West. Military uses mils for precision. Compass has 32 points (11.25° each). GPS uses decimal degrees.

• Bearings: 0-360° from North
• NATO mil: 6400 per circle
• Compass points: 32 (11.25° each)
• GPS: decimal degrees

Star positions: arcseconds precision. Parallax: milliarcseconds. Hubble: ~50 mas resolution. Gaia satellite: microarcsecond precision. Hour angle: 24h = 360°.

• Arcsecond: star positions
• Milliarcsecond: parallax, VLBI
• Microarcsecond: Gaia satellite
• Hour angle: 15°/hour

Slope: percent grade or angle. 10% grade ≈ 5.7°. Road design uses percent. Surveying uses degrees/minutes/seconds. Gradian system for metric countries.

• Slope: % or degrees
• 10% ≈ 5.7° (arctan 0.1)
• Surveying: DMS (deg-min-sec)
• Gradian: metric surveying

rad = deg × π/180. deg = rad × 180/π. Quick: 180° = π rad, so divide/multiply by this ratio.

• rad = deg × 0.01745
• deg = rad × 57.2958
• π rad = 180° (exact)
• 2π rad = 360° (exact)

angle = arctan(slope/100). 10% slope = arctan(0.1) ≈ 5.71°. Reverse: slope = tan(angle) × 100.

• θ = arctan(grade/100)
• 10% → arctan(0.1) = 5.71°
• 45° → tan(45°) = 100%
• Steep: 100% = 45°

1° = 60′ (arcmin). 1′ = 60″ (arcsec). Total: 1° = 3600″. Quick subdivision for precision.

• 1° = 60 arcminutes
• 1′ = 60 arcseconds
• 1° = 3600 arcseconds
• DMS: degrees-minutes-seconds

Road has 8% grade. What's the angle?

θ = arctan(8/100) = arctan(0.08) ≈ 4.57°. Relatively gentle slope!

Navigate 135° bearing. What compass direction?

0°=N, 90°=E, 180°=S, 270°=W. 135° is between E (90°) and S (180°). Direction: Southeast (SE).

Star moved 0.5 arcseconds. How many degrees?

1″ = 1/3600°. So 0.5″ = 0.5/3600 = 0.000139°. Tiny movement!

Babylonians used base-60 (sexagesimal) system. 360 has many divisors (24 factors!). Approximately matches 360 days in year. Convenient for astronomy and timekeeping. Also divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12...

Radian defined by arc length = radius. Makes calculus beautiful: d/dx(sin x) = cos x (only in radians!). In degrees, d/dx(sin x) = (π/180)cos x (messy). Nature 'uses' radians!

Metric angle: 100 grad = right angle. Tried during French Revolution with metric system. Never popular—degrees too entrenched. Still used in some surveying (Switzerland, northern Europe). Calculators have 'grad' mode!

1 milliarcsecond ≈ width of human hair viewed from 10 km away! Hubble Space Telescope can resolve ~50 mas. Incredible precision for astronomy. Used to measure stellar parallax, binary stars.

Military mil: 1 mil ≈ 1 m width at 1 km distance (NATO: 1.02 m, close enough). Easy mental math for range estimation. Different countries use different mils (6000, 6300, 6400 per circle). Practical ballistics unit!

90 = 360/4 (quarter turn). But 'right' comes from Latin 'rectus' = upright, straight. Right angle makes perpendicular lines. Essential for construction—buildings need right angles to stand!

The Babylonians used a sexagesimal (base-60) number system for astronomy and timekeeping. They divided the circle into 360 parts because 360 ≈ days in a year (actually 365.25), and 360 has 24 divisors—incredibly convenient for fractions.

This base-60 system persists today: 60 seconds per minute, 60 minutes per hour and per degree. The number 360 factors as 2³ × 3² × 5, dividing evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180—a calculator's dream!

• 2000 BCE: Babylonian astronomers track celestial positions in degrees
• 360° chosen for divisibility and ~year approximation
• Base-60 gives us hours (24 = 360/15) and minutes/seconds
• Greek astronomers adopt 360° from Babylonian tables

Euclid's Elements (300 BCE) formalized angle geometry—right angles (90°), complementary (sum to 90°), supplementary (sum to 180°). Greek mathematicians like Hipparchus created trigonometry using degree-based tables for astronomy and surveying.

Medieval navigators used the astrolabe and compass with 32 points (each 11.25°). Mariners needed precise bearings; arcminutes (1/60°) and arcseconds (1/3600°) emerged for star catalogs and nautical charts.

• 300 BCE: Euclid's Elements defines geometric angles
• 150 BCE: Hipparchus creates first trig tables (degrees)
• 1200s: Astrolabe uses degree markings for celestial navigation
• 1569: Mercator map projection requires angle-preserving math

As Newton and Leibniz developed calculus (1670s), degrees became problematic: d/dx(sin x) = (π/180)cos x in degrees—an ugly constant! Roger Cotes (1682-1716) and Leonhard Euler formalized the radian: angle = arc length / radius. Now d/dx(sin x) = cos x beautifully.

James Thomson coined 'radian' in 1873 (from Latin 'radius'). The radian became THE unit for mathematical analysis, physics, and engineering. Yet degrees persisted in everyday life because humans prefer whole numbers over π.

• 1670s: Calculus reveals degrees create messy formulas
• 1714: Roger Cotes develops 'circular measure' (pre-radian)
• 1748: Euler uses radians extensively in analysis
• 1873: Thomson names it 'radian'; becomes math standard

WWI artillery demanded practical angle units: the mil was born—1 mil ≈ 1 meter deviation at 1 km distance. NATO standardized 6400 mils/circle (nice power of 2), while USSR used 6000 (decimal convenience). True milliradian = 6283/circle.

Space-age astronomy achieved milliarcsecond precision (Hipparcos, 1989), then microarcseconds (Gaia, 2013). Gaia measures stellar parallax to 20 microarcseconds—equivalent to seeing a human hair from 1,000 km away! Modern physics uses radians universally; only navigation and construction still favor degrees.

• 1916: Military artillery adopts mil for range calculations
• 1960: SI recognizes radian as coherent derived unit
• 1989: Hipparcos satellite: ~1 milliarcsecond precision
• 2013: Gaia satellite: 20 microarcsecond precision—maps 1 billion stars

Use degrees for: everyday angles, navigation, surveying, construction. Use radians for: calculus, physics equations, programming (trig functions). Radians are 'natural' because arc length = radius × angle. Derivatives like d/dx(sin x) = cos x only work in radians!

Circle circumference = 2πr. Half circle (straight line) = πr. Radian defined as arc length/radius. For half circle: arc = πr, radius = r, so angle = πr/r = π radians. Therefore π rad = 180° by definition.

Use arctan: angle = arctan(grade/100). Example: 10% grade = arctan(0.1) ≈ 5.71°. NOT just multiply! 10% ≠ 10°. Reverse: grade = tan(angle) × 100. 45° = tan(45°) × 100 = 100% grade.

Arcminute (′) = 1/60 of a degree (angle). Minute of time = 1/60 of an hour (time). Completely different! In astronomy, 'minute of time' converts to angle: 1 min = 15 arcminutes (because 24h = 360°, so 1 min = 360°/1440 = 0.25° = 15′).

Mil designed so 1 mil ≈ 1 meter at 1 km (practical ballistics). True mathematical milliradian = 1/1000 rad ≈ 6283 per circle. NATO simplified to 6400 (power of 2, divides nicely). USSR used 6000 (divides by 10). Sweden 6300 (compromise). All close to 2π×1000.

Yes! Positive = counterclockwise (math convention). Negative = clockwise. -90° = 270° (same position, different direction). In navigation, use 0-360° range. In math/physics, negative angles are common. Example: -π/2 = -90° = 270°.