Degrees to Radians Converter

Radians are the natural unit of angle for mathematics. One radian is the angle subtended by an arc whose length equals the circle's radius, so radians make arc length s = r * theta a one-line identity instead of a formula stuffed with conversion factors. The bigger payoff is in calculus: the derivative of sin x is cos x only when x is in radians. If you measure x in degrees, the derivative becomes (pi/180) * cos x, and that extra constant propagates through every Taylor series, every Fourier expansion, and every differential equation in physics. Engineers and scientists treat radians as the default for the same reason chemists default to moles: the unit makes the equations cleaner.

By definition, one radian is the angle at the center of a circle that cuts off an arc equal to one radius. The circumference of a circle is C = 2 * pi * r, so the total arc all the way around contains 2*pi radii laid end to end. That is the full turn, the same angle we call 360 degrees. The two scales are pinned together by 2*pi rad = 360 deg, which is why the converter just multiplies by pi/180 in one direction and 180/pi in the other. Half a turn is pi rad = 180 deg, a quarter turn is pi/2 rad = 90 deg, and so on through the table of common angles.

It is inherited from Babylonian astronomers around 2000 to 1500 BCE, who used a base-60 (sexagesimal) number system and rounded the solar year to a convenient 360 days. Dividing the sun's apparent yearly path into 360 equal steps gave one degree per day, near enough. Base-60 was attractive because 60 has more small divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, 30) than any smaller number, so degrees split cleanly into halves, thirds, quarters, fifths, sixths, and so on, all the way down to the 30-60-90 and 45-45-90 special triangles still drilled in geometry class. We kept the convention long after we stopped tracking the calendar with it.

Use degrees when a human will read the value: navigation, surveying, machining, CAD, drafting, weather, latitude and longitude, and any context where the original measurement was taken with a protractor or compass. Use radians when an equation will read the value: every calculus class, every physics formula with sin, cos, or tan (simple harmonic motion, wave equations, AC circuits, rotational dynamics), and every programming language's standard library, since Math.sin, Math.cos, and Math.tan in JavaScript, Python, C, and most others take radians by default. The cleanest workflow is to enter measured values in degrees, convert to radians for the math, then convert any final angle back to degrees for the report.

The gradian (also called gon) is the metric attempt at angle measurement. One full turn is 400 grad, so a right angle is exactly 100 grad and percentages of a turn map cleanly to gradians (25% of a turn = 100 grad). It was promoted alongside the meter, kilogram, and liter during the French Revolution to push base-10 thinking into every unit of measurement. Surveying calculators, especially in continental Europe, still expose a grad mode, and the unit shows up occasionally in mining and artillery. Almost nowhere else: trigonometry textbooks use degrees and radians, scientific work uses radians, and consumer-grade calculators offer grad mostly as a third option you switch on by accident. The converter reports gradians for completeness, not because you will need them often.