Rotation Calculator - Rotate Points in 2D

Your details

Counterclockwise is the standard math convention. Clockwise negates the angle before applying the formula.
Pick a standard angle to pre-fill the angle field, or choose Custom to enter any value.
The rotation angle. Positive values rotate counterclockwise; use Clockwise direction to flip the sign.
deg
The x-coordinate of the point you want to rotate.
The y-coordinate of the point you want to rotate.
New X (x')
-4

The x-coordinate of the rotated point

New Y (y')3
Distance from center5
Angle (radians)1.570796
Angle (degrees)90
Initial angle from center53.1301deg
Final angle from center143.1301deg
New X (x')-4
New Y (y')3

(3, 4) rotated 90.0000 deg counterclockwise becomes (-4.000000, 3.000000).

  • The point stays 5.0000 units from the center of rotation. Rotation is an isometric transformation: it preserves all distances.
  • The initial angle from the center was 53.13 deg and the final angle is 143.13 deg, a difference of 90.0000 deg.
  • A 90-degree rotation swaps and negates coordinates: (x, y) becomes (-y, x) around the origin.
  • The rotation matrix [[cos t, -sin t], [sin t, cos t]] encodes this transformation compactly and can be applied to any number of points by matrix multiplication.

Next stepTo rotate a shape, apply the same formula to every vertex. The shape will be congruent to the original: same side lengths, same angles, just repositioned.

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