How 2D Point Rotation Works
Rotating a point in a 2D plane means moving it along a circular arc around a fixed center point, through a given angle, without changing its distance from that center. The standard tool for this is the 2D rotation matrix: for a point (x, y) rotated by angle θ counterclockwise around the origin, the new coordinates are x′ = x·cosθ − y·sinθ and y′ = x·sinθ + y·cosθ. When the center of rotation is not the origin, you first shift the point so the center sits at the origin, apply the same rotation, then shift back: x′ = cx + (x−cx)cosθ − (y−cy)sinθ and y′ = cy + (x−cx)sinθ + (y−cy)cosθ, where (cx, cy) is the pivot point.
Formula and method
Enter the point's (x, y) coordinates, the center of rotation (cx, cy) — use (0, 0) to rotate around the origin — the rotation angle, and whether the angle is in degrees or radians. If you enter degrees, the calculator first converts θ to radians (θ_rad = θ_deg × π/180) since JavaScript's trigonometric functions expect radians. Choosing "clockwise" simply negates the angle before it is applied, since clockwise rotation is the mirror of counterclockwise rotation in the standard x-right, y-up coordinate system. The calculator then applies the rotation-matrix formula above to compute the new coordinates (x′, y′).
Common sources of error
- Degrees vs. radians: a 90 typed into a field expecting radians rotates the point by roughly 5,157° instead of 90° — always confirm the angle unit selector matches your input.
- Wrong center point: forgetting to set the center of rotation (leaving it at a stale value) rotates the point around the wrong pivot, giving a plausible-looking but incorrect answer.
- Direction confusion: mixing up clockwise and counterclockwise flips the sign of the angle and lands the point in the wrong quadrant — check the sign convention before trusting the result.
Checking your result
Because rotation is a rigid transformation, the distance from the center to the rotated point must exactly equal the distance from the center to the original point — this calculator shows that distance so you can confirm it did not change. As a quick sanity check, a 90° counterclockwise rotation of a point on the positive x-axis relative to the center should land on the positive y-axis relative to that same center, and a 180° rotation should land diametrically opposite the original point.
Applications
Point rotation is fundamental to computer graphics and game development (rotating sprites, models, and cameras), robotics and CNC machining (reorienting tool paths and coordinate frames), CAD and engineering drawings (rotating shapes around a pivot), and physics problems involving circular or rotational motion. In every case, the same rotation-matrix formula applies — only the center point, angle, and coordinate convention change.