Coterminal Angle Calculator
Reduce an angle to an equivalent angle from zero up to but excluding 360 degrees.
How the result changes
Only angle changes; all other inputs stay at your current values. The line illustrates the formula across the shown range. This is not a forecast of real-world outcomes.
About this calculation
Coterminal Angle Calculator: Reduce an angle to an equivalent angle from zero up to but excluding 360 degrees. Enter angle; the reported quantities are principal coterminal angle (degrees), full turns removed (turns). The equations and examples below show how those inputs lead to the result and where the model's assumptions matter.
Formula
principal angle = ((angle mod 360) + 360) mod 360
Worked example
With Angle: 450 degrees, principal coterminal angle is 90 degrees.
Method and inputs
Adding or subtracting any whole multiple of 360 degrees leaves the terminal side unchanged. Apply a modulo operation twice so negative inputs become a nonnegative principal angle. The second output counts the signed full turns removed.
Worked example
For 450 degrees, subtract 360 to obtain the principal coterminal angle of 90 degrees; one full positive turn was removed. For -90 degrees, adding 360 yields 270 degrees and the turn count is -1.
A clockwise turn of -450 degrees finishes at 270 degrees on the unit circle, just like -90 degrees. The principal-angle output discards the extra full revolution, but the signed turn count retains it. This matters in animation and motion control where path history matters as much as final orientation.
How to read the result
Coterminal angles have the same direction on a unit circle but can record different amounts of rotation. A principal value is convenient for trigonometric lookup, whereas the original angle can matter when counting physical revolutions.
Use the principal result to compare directions, not total rotations. A rotor at 45 degrees and one at 765 degrees point alike, yet the second has completed two extra turns. The two numerical outputs deliberately preserve these different interpretations.
Assumptions and checks
The input and outputs are in degrees. This method does not accept radians directly; multiply a radian value by 180/pi first. Extremely large angles can lose precision in ordinary floating-point arithmetic.
A negative remainder in programming is not always the principal angle. The double-modulo step corrects that. Do not subtract 180 degrees, because that points in the opposite direction rather than returning to the same terminal side.
Common questions
What does the coterminal angle calculator calculate?
It finds the angle in one standard 360-degree revolution that has the same terminal ray as the entered rotation, together with the number of full turns removed.
When should I recalculate or check another method?
Recalculate after converting any radian measure to degrees. If signed rotation direction is needed for a motor or animation, retain the original angle as well as the principal equivalent.
Inputs
- Angle (degrees)