How to use the complex plane rotation calculator
On the complex plane a point (x, y) is the complex number z = x + yi. Multiplying it by r(cos θ + i sin θ) rotates it counterclockwise by θ and scales its length by r. Enter the point, the angle θ and the factor r, and the tool returns the rotated point together with the multiplier, the modulus and the argument.
The formulas used are x′ = r(x cos θ − y sin θ) and y′ = r(x sin θ + y cos θ). If you give a center of rotation, the point is shifted so the center becomes the origin, rotated, and shifted back, which is exactly what you need when a figure turns about one of its own vertices. Leave the center blank to rotate about the origin.
The argument comes from atan2 and is reported in (−180°, 180°], because the arctangent of y/x alone cannot tell the second quadrant from the fourth. When the point coincides with the center it has no direction, so its argument is undefined; those rows are hidden and a note appears instead. The factor r must be greater than 0.
Angles beyond 360° are accepted, and a negative angle rotates clockwise. The result also folds the angle you entered into the (−180°, 180°] range. Irrational values such as cos 45° are printed to 6 decimal places, and a value is treated as zero when its absolute value is below 1e-9.
Frequently asked questions
Complex multiplication multiplies the moduli and adds the arguments. Multiplying by a number of modulus r and argument θ therefore scales the length by r and turns the direction by θ.
Yes. Enter a negative angle. For example −90° is the same as a 90° clockwise rotation.
Yes. Enter the coordinates of the center and the point turns about it. The tool shifts the center to the origin, rotates, then shifts back.