Angle Converter

About angle units

Angle measures the rotation between two rays sharing a common endpoint. The degree is the everyday unit: a full circle is 360°, which descends from ancient Babylonian base-60 arithmetic and the rough approximation of 360 days in a year. The radian is the SI unit used in mathematics and physics: one radian is the angle subtended at the center of a circle when the arc length equals the radius, so a full circle spans exactly 2π radians (≈ 6.2832 rad). The gradian (or gon) divides a right angle into 100 parts, giving 400 gradians to a full circle — a system designed for decimal arithmetic and still used in surveying and some civil-engineering applications.

History & origin

The division of a circle into 360 degrees traces to Babylonian astronomy around 1000 BCE, likely chosen because 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 12 — making fractional degrees easy in a base-60 number system. The radian was formalised in the 19th century, with the name attributed to mathematician James Thomson around 1873. The gradian was introduced during the French Revolutionary metric reforms of the 1790s alongside meters and liters, though it never achieved the same global adoption.

Tips

Radians catch people off guard in programming because most calculators default to degrees, but every programming language's trigonometric functions (sin, cos, tan) expect radians. Passing degrees directly into sin() produces wrong results. The conversion is straightforward: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. A right angle is 90° = π/2 rad ≈ 1.5708 rad; this benchmark is worth memorising as a sanity check.

Common questions

Why are there 360 degrees in a circle?

The convention comes from ancient Babylonian astronomy, which used a base-60 (sexagesimal) number system. 360 was a convenient approximation for the number of days in a year and is highly composite — divisible by 24 different integers — making it easy to subdivide circles into equal parts without fractions.

What is a gradian, and who uses it?

A gradian (also called a gon) is 1/400 of a full circle, so a right angle equals exactly 100 gradians. The unit was designed for decimal arithmetic in surveying, where compass bearings and slope angles are often calculated together. It is common in European civil engineering and land surveying but rare elsewhere.