Angle Converter Guide: Degrees, Radians, and Gradians

Degrees feel natural because we grew up with them, but radians are what most math and physics actually run on, and gradians show up in surveying and some European engineering contexts. Mixing these up in a formula is one of the most common sources of wrong answers in trigonometry homework and code alike.

The Three Systems and Their Full Circle

A full rotation is 360 degrees, 2*pi radians (about 6.2832 radians), or 400 gradians. Degrees split the circle into 360 arbitrary parts, a convention traced back to ancient Babylonian base-60 counting. Radians define an angle by the ratio of arc length to radius, making one radian equal to about 57.2958 degrees. Gradians divide the circle into 400 units so that a right angle is a clean 100 gradians, which surveyors liked for decimal-based calculations.

Converting Between Degrees and Radians

The formula is radians = degrees times (pi divided by 180), and degrees = radians times (180 divided by pi). A 90-degree right angle equals pi/2 radians, roughly 1.5708. A 45-degree angle equals pi/4, about 0.7854 radians. To go the other direction, 1 radian equals about 57.2958 degrees, so 2 radians is roughly 114.59 degrees. Programming languages like JavaScript, Python, and C all expect radians in their built-in trig functions (Math.sin, Math.cos), which is why a common bug is passing in degrees directly and getting wrong results.

Working With Gradians

Gradians (also called gons) convert with gradians = degrees times (400/360), or simplified, degrees times (10/9). So 90 degrees equals 100 gradians, and 180 degrees equals 200 gradians. To convert gradians to radians, multiply by pi/200. Gradians rarely appear outside of certain surveying instruments, some scientific calculators (often labeled "GRAD" mode), and French engineering traditions, but if you inherit old land survey data, you may run into them.

A Worked Example: Roof Pitch to Angle

Say a roof rises 6 inches for every 12 inches of horizontal run, a "6-in-12" pitch. The angle from horizontal is arctan(6/12) = arctan(0.5), which comes out to about 26.57 degrees. In radians that is about 0.4636. If a spec sheet demands gradians for some reason, multiply the degree value by 10/9 to get roughly 29.52 gradians. Getting the unit wrong here could mean a contractor cuts rafters at the wrong angle entirely.

Common Conversion Mistakes

The most frequent error is forgetting that most programming and spreadsheet trig functions expect radians, not degrees. Excel's SIN, COS, and TAN functions all take radians, so you need RADIANS(45) to get the right input for a 45-degree angle; skipping that conversion is a classic spreadsheet bug. Another mistake is truncating pi too early — using 3.14 instead of a fuller value like 3.14159265 can introduce visible error in engineering-scale calculations, especially over repeated operations.

Reference Table for Quick Lookups

• 30 degrees = 0.5236 radians = 33.33 gradians
• 45 degrees = 0.7854 radians = 50 gradians
• 60 degrees = 1.0472 radians = 66.67 gradians
• 90 degrees = 1.5708 radians = 100 gradians
• 180 degrees = 3.1416 radians = 200 gradians
• 360 degrees = 6.2832 radians = 400 gradians

Convert Angles Instantly

Our angle converter switches between degrees, radians, and gradians in real time, so you never have to remember whether to multiply by pi/180 or 180/pi. Enter a value in any unit and see the other two update automatically.