How to rotate a coordinate point
Use the calculator
Enter x and y for one point, choose degrees or radians, and enter the signed rotation angle. Select Rotate point to calculate x′ and y′. Any edit clears the previous result and disables meaningful copying until the next calculation.
This page always rotates around (0, 0). Positive is counterclockwise in a standard Cartesian plane; use a negative angle for clockwise rotation.
Rotation formulas
For angle θ, x′ = x cos θ − y sin θ and y′ = x sin θ + y cos θ. The same formula handles positive, negative and multiple-turn angles.
Rotation is rigid: √(x′² + y′²) equals √(x² + y²), apart from ordinary floating-point effects. This preserved radius is shown as a quick verification.
Scope and limitations
The calculator rotates one 2D point around the origin. It does not use an arbitrary pivot, transform an entire shape, edit a matrix or perform 3D rotations.
Ordinary results show at most two decimal places, while the browser number is retained until display. Near-zero sine or cosine noise is cleaned so quarter-turn examples read naturally.
x′ = x cos θ − y sin θ · y′ = x sin θ + y cos θ
Worked examples
90° and 180°
Rotating (3, 4) by 90° gives (−4, 3). Rotating the same point by 180° gives (−3, −4).
360°, zero and negative angles
A 360° turn returns the original point. The origin remains (0, 0) for every angle. Rotating (1, 0) by −90° gives (0, −1).
Frequently asked questions
Which direction is positive?
Positive angles are counterclockwise in the usual x-right, y-up coordinate plane.
Does rotation change distance from the origin?
No. A pure rotation preserves the radius exactly in the mathematical model.
Can I rotate around another point?
Not on this page. Translate to that center, rotate, and translate back, or use a dedicated pivot tool.