Degrees vs radians: when to use each angle unit in math

Degrees and radians measure the same thing — the amount of rotation between two rays — using two different reference points. A degree divides a full turn into 360 equal slices, a convention inherited from Babylonian sexagesimal astronomy. A radian instead ties the angle to geometry directly: it's the angle at the center of a circle subtended by an arc whose length equals the circle's radius. Both describe the same physical angle; they just count it differently.

The exact conversion

Because a full circle is 360° and also exactly 2π radians, one radian equals 180/π degrees — approximately 57.2957795131 degrees. Going the other way, one degree equals π/180 radians, approximately 0.0174532925 radians.

90° × (π / 180) = 1.5708 rad
1 rad × (180 / π) = 57.2958°

Where each unit is actually used

Degrees dominate everyday and applied contexts: compass headings, latitude and longitude, construction and framing angles, protractors, and consumer-facing tools. Radians dominate mathematics, physics, and programming: calculus identities involving trigonometric functions are only clean in radians, and functions like Math.sin() in JavaScript or math.sin() in Python expect their argument in radians, not degrees.

A common bug: mixing units in one formula

The single most common angle-related bug in code and spreadsheets is passing a degree value to a function that expects radians (or vice versa) — the calculation runs without error but returns a nonsensical result, since sin(90) in radians mode is very different from sin(90°). Spreadsheet functions like Excel's SIN() also expect radians, which is why Excel provides a separate RADIANS() helper function to convert a degree value first. When a trig result looks obviously wrong, checking whether the input unit matches what the function expects is the first thing worth verifying.