Angle Converter

Convert between degrees, radians, and gradians, with why a circle has 360 degrees explained.

Why a Circle Has 360 Degrees, and What a Radian Actually Is

The 360-degree circle is another inheritance from ancient Babylonian mathematics, likely chosen because 360 divides evenly by an unusually large number of whole numbers, and possibly because it sits close to the number of days in a year as understood by early astronomers tracking the sun’s yearly path, making it a convenient number for dividing up a circle into clean fractions for both mathematics and astronomy. The radian works on entirely different logic, defined not by arbitrary convention but by the actual geometry of a circle itself, specifically as the angle created when the arc length along a circle’s edge equals its radius, which means a full circle contains exactly two pi radians, an irrational number that makes radians look awkward in everyday conversation but genuinely elegant in mathematics and physics, where formulas involving angles become simpler and more natural when expressed in radians rather than degrees. The gradian, far less commonly used today, was a deliberate attempt during the same era as the metric system’s creation to bring angle measurement onto a clean base-ten footing, dividing a full circle into 400 gradians so that a right angle equals exactly 100, and while it never achieved wide adoption outside surveying and some European engineering contexts, it remains a genuinely useful illustration of how differently an angle can be divided depending on which property, whole-number divisibility, geometric elegance, or decimal convenience, a system is optimized around.