Convert geometric angles for math, physics, and design.
Angles measure the amount of rotation between two intersecting lines. They are a fundamental concept in geometry, trigonometry, physics, and engineering. While most people are familiar with degrees from grade school protractors, higher-level mathematics and specific fields use different units like radians and gradians.
The degree is the most common unit of angle measurement in everyday life. A full circle is divided into 360 degrees. The historical reason for using 360 is traced back to the ancient Babylonians, whose calendar year was roughly 360 days, and the fact that 360 is highly divisible by many numbers.
The radian is the standard unit of angular measure in mathematics, particularly in calculus. A radian is defined by the length of the arc of a circle. When the arc length equals the radius of the circle, the angle formed is exactly 1 radian. A full circle is equal to 2π radians (roughly 6.283 radians).
Also known as the gon or grad, a gradian divides a full circle into 400 parts. This system was introduced alongside the metric system in France to align angle measurement with base-10 decimals (making a right angle exactly 100 gradians). While it never replaced the degree entirely, it is still used in some areas of surveying and civil engineering in Europe.
If you need to perform calculations manually or write code, here are the core relationships based on a full circle:
360 Degrees = 2π Radians = 400 Gradians
Degrees to Radians: multiply by (π / 180)
Radians to Degrees: multiply by (180 / π)