Rotation Calculator (rotate a point by an angle about the origin or any centre)

Understanding the point rotation calculator — concept and principle

This calculator rotates a point (x, y) by any angle, clockwise or counterclockwise, about the origin or about any centre you choose. It gives the new coordinates, the unchanged distance from the centre, the equivalent counterclockwise angle and the cosine and sine used. Use it for coordinate geometry, transformations homework, computer graphics, CAD and robotics.

How it is calculated

Rotation counterclockwise by θ about the centre (a, b):

x′ = a + (x − a)·cos θ − (y − b)·sin θ

y′ = b + (x − a)·sin θ + (y − b)·cos θ

• About the origin, a = b = 0.
• A clockwise turn by θ is a counterclockwise turn by −θ.
• Multiples of 90° use exact cosine and sine values, so results have no rounding dust.

The principle

The rotation matrix [cos θ −sin θ; sin θ cos θ] sends the x-axis direction (1, 0) to (cos θ, sin θ) and the y-axis direction (0, 1) to (−sin θ, cos θ). Every other point is a combination of those two directions, so it rotates the same way. To rotate about a different centre, you move the centre to the origin, rotate, and move back. Rotation is a rigid motion: it keeps distances and angles, so the point stays on the same circle around the centre.

Worked example

Default inputs: point (5, 2), θ = 90°, counterclockwise, centre (1, 1).

1. Shift so the centre is at the origin: (5 − 1, 2 − 1) = (4, 1).
2. cos 90° = 0 and sin 90° = 1.
3. Rotate: (4 × 0 − 1 × 1, 4 × 1 + 1 × 0) = (−1, 4).
4. Shift back: (−1 + 1, 4 + 1) = (0, 5).
5. Distance from the centre: √(4² + 1²) = √17 ≈ 4.12311, before and after.

How to read the result

The new coordinates are where the point lands. The distance from the centre should be the same before and after; that is a quick check. A 90° counterclockwise turn about the origin maps (x, y) to (−y, x), 180° to (−x, −y) and 270° to (y, −x).

Common mistakes

• Rotating about the origin when the problem says to rotate about another point.
• Mixing up clockwise and counterclockwise; in mathematics, positive angles are counterclockwise.
• Entering radians when the field expects degrees.
• In screen coordinates, where y increases downward, the visual direction appears reversed.

Frequently asked questions

What is the rule for a 90° rotation about the origin?

Counterclockwise, (x, y) becomes (−y, x); clockwise, it becomes (y, −x).

Does rotation change the shape of a figure?

No. Rotating every vertex keeps all lengths and angles.

How do I rotate a whole shape?

Rotate each vertex with the same angle and centre, then join them in the same order.

Sources: OpenStax, Precalculus 2e (2021), §10.4 Rotation of Axes