Angle Converter

Angles measure the amount of rotation between two lines meeting at a point. The most familiar unit is the degree (°), where a full circle is divided into 360 equal parts. The number 360 was chosen by ancient Babylonians — it is close to the number of days in a year, and it has an unusually large number of integer divisors (24 of them), making it easy to divide a circle into halves, thirds, quarters, sixths, eighths, and more with whole numbers.

Radians are the SI unit of angle measurement and are used universally in mathematics and physics. One radian is defined as the angle where the arc length equals the radius of the circle. Since a circle's circumference is 2πr, a full circle equals exactly 2π radians ≈ 6.2832 radians. Radians make trigonometric formulas far simpler: the Taylor series for sin(x) is x − x³/6 + x⁵/120 − ..., but only when x is in radians. Using degrees would require a conversion factor in every formula, which is why pure mathematics and physics always use radians.

Gradians (also called gon or grad) divide a right angle into 100 parts, making a full circle exactly 400 gradians. They were introduced during the French Revolution alongside the metric system and are still used in surveying, geodesy, and civil engineering in parts of Europe. A slope of 1% corresponds to 1 gradian, making them practical for gradient measurements.

Arcminutes (1/60 of a degree) and arcseconds (1/3,600 of a degree) are used in astronomy, navigation, and optometry. Latitude and longitude coordinates are specified in degrees, minutes, and seconds. A GPS receiver provides position to within a few meters, corresponding to milliarcseconds of accuracy. The angular diameter of the full Moon as seen from Earth is approximately 31 arcminutes.