Degrees ↔ Radians Interactive Protractor

The degree (360 parts per circle) comes from Babylonian astronomers around 2000 BC. They chose 360 because their year had ~360 days and because 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), making fractional calculations tractable in a base-60 (sexagesimal) number system. Ptolemy's Almagest (c. 150 AD) cemented the 360° circle for celestial calculation, and the convention has survived three millennia largely because of the divisibility advantage and entrenched maps/instruments.

The radian is much younger. Roger Cotes (1682-1716), an English mathematician and Newton's editor on the Principia, wrote in his posthumously published Harmonia Mensurarum (1722) that the natural unit of angle is one where the arc length equals the radius — what he called the "mensura unius". The word "radian" itself was coined by James Thomson (brother of Lord Kelvin) in 1873 examination papers at Queen's College Belfast. By the end of the 19th century, radians had become the standard angle unit in calculus and physics.

The radian's power is in calculus. The Taylor series for sin and cos converge only when the argument is in radians: sin(x) = x − x³/6 + x⁵/120 − ... requires x in radians for the series to actually equal sin(x). The derivative of sin(x) is cos(x) only when x is in radians; in degree-mode you pick up a π/180 factor every time you differentiate. Every physics formula involving circular motion (ω = 2πf, v = rω, a = ω²r) requires radians for unit consistency.

Modern computing libraries universally accept radians. JavaScript Math.sin(x) takes x in radians; Python math.sin too; C's sin() too. Most languages provide degrees-to-radians conversions as separate helper functions. Spreadsheets are split: Excel and Google Sheets have SIN() requiring radians, plus separate DEGREES() and RADIANS() conversion helpers. Most scientific calculators have a DEG/RAD/GRAD mode switch — getting this wrong is the #1 trigonometry homework error.

The special angles in the unit circle (30°, 45°, 60°, 90° and their reflections) appear everywhere because their sines and cosines are simple radicals. sin(30°) = sin(π/6) = 1/2. sin(45°) = sin(π/4) = √2/2. sin(60°) = sin(π/3) = √3/2. sin(90°) = sin(π/2) = 1. These come from 30-60-90 and 45-45-90 right triangles built from the equilateral triangle and the square — geometry that pre-dates trigonometry. Modern phone screens, CAD programs, and game engines have hardware support for these exact values via SIMD instructions.

The gradian (gon) divides the circle into 400 parts; 100 gons in a right angle. Invented during the French Revolution alongside the metric system, it survives in European surveying — French and German cadastral records often use gons. Civil engineering software supports it. 1 gon = 0.9° = π/200 rad. The gon never displaced the degree because the 360 system was too entrenched in instruments and tables, and the radian had already taken over the science community.

Every conversion in this tool uses the exact factor π/180 stored to IEEE 754 double-precision (~15 significant decimal digits). The π-fraction display recognizes all 22 special angles up to 360° and prints "π/4" rather than the decimal 0.7853981633974483. This matters in symbolic-math workflows (Mathematica, SymPy, Maxima) and in teaching contexts where the rational structure of radians is the point.