Radians vs Degrees
Degrees and radians describe the same angle idea in different units. Degrees divide a full turn into 360 parts, while radians connect angle measure directly to circle radius and arc length.
Key relationship
180 deg = pi rad
One full turn is 360 degrees or 2pi radians.
When degrees and radians are useful
Use degrees for diagrams
Degrees are familiar for geometry drawings, protractors, bearings, and introductory angle measurement.
Use radians for unit circle work
Radians connect angle measure to radius, arc length, calculus, and exact unit circle notation.
Convert through pi
Multiply degrees by pi over 180, or multiply radians by 180 over pi.
How to move between angle units
A full turn is 360 degrees. The same full turn is 2pi radians. That means half a turn is 180 degrees or pi radians, and a quarter turn is 90 degrees or pi over 2 radians. Most classroom conversions come from this relationship.
To convert degrees to radians, multiply by pi over 180. To convert radians to degrees, multiply by 180 over pi. Exact unit circle angles are often written as pi multiples because they are more precise and easier to connect to sine, cosine, and tangent values than rounded decimals.
Degrees are not wrong or less useful. They are often the best choice for drawings and everyday angle descriptions. Radians become more natural when formulas involve arc length, angular speed, calculus, or the unit circle. Toolarithm includes both the converter and the unit circle reference so users can move between the notations without losing the meaning of the angle.