Free Rotation Calculator

Last updated October 5, 2026 · Report an error

Thanks for rating!

Rotating a point in 2D turns it through a given angle around a fixed center, tracing an arc of a circle. This is one of the fundamental "rigid transformations" in geometry — it preserves distances and shapes while changing orientation — and it underlies computer graphics (rotating a sprite or 3D model on screen), robotics (computing a rotated joint or end-effector position), and CAD software (rotating a design element around a pivot point). When the center isn't the origin, the calculator handles that by first shifting the whole picture so the center sits at the origin, applying the standard rotation matrix, then shifting back.

How it works

Enter the point, an optional center (default the origin), and the rotation angle. The calculator translates the point so the center of rotation becomes the origin (subtracting the center's coordinates), applies the standard 2D rotation matrix, x'=x cosθ-y sinθ and y'=x sinθ+y cosθ, then translates the result back by adding the center's coordinates again.

Enter point — x.
Enter point — y.
Enter center of rotation — x (default 0).
Enter center of rotation — y (default 0).
Enter rotation angle (counterclockwise).
Enter angle unit.
Click Calculate to see your results.

Examples

(1,0) rotated 90° about the origin

x'=1·cos90°-0·sin90°=1(0)-0(1)=0. y'=1·sin90°+0·cos90°=1(1)+0(0)=1. Result: (0, 1) — a quarter-turn counterclockwise.

(3,3) rotated 180° about center (2,2)

Translate: (3-2, 3-2)=(1,1). Rotate 180°: x'=1(-1)-1(0)=-1, y'=1(0)+1(-1)=-1, giving (-1,-1). Translate back: (-1+2, -1+2)=(1,1). The result is (1,1) — the point directly opposite (3,3) through the center (2,2).

Common mistakes to avoid

Forgetting to translate back to the original center after rotating around a non-origin center.
Rotating clockwise when a positive angle was intended, or vice versa — double-check the sign convention.
Using degrees when the underlying formula expects radians (or vice versa) without converting first.

Frequently asked questions

Counterclockwise for a positive angle, matching the standard mathematical convention (the same one used for measuring angles on the unit circle). A negative angle rotates clockwise.

No — rotation is a "rigid" transformation, so the distance from the rotated point back to the center of rotation is always exactly the same as it was before rotating.

The point returns exactly to its starting position, since 360° is one full revolution.

Rotating about the origin uses the rotation matrix directly. Rotating about any other center requires first "recentering" the point relative to that center, rotating, then shifting back — otherwise the point would swing around the wrong pivot.

Yes, in general — only the distance to the specific center of rotation is preserved; distances to any other fixed point typically change as the point sweeps around its circular arc.

2D rotation always happens around a single point using one angle; 3D rotation instead happens around an axis (a whole line, not just a point) and generally requires a more involved rotation matrix depending on which axis is chosen.