# Where do points go in a rotation?

Rotates points in the coordinate plane by any angle, counterclockwise or clockwise, about the origin or any centre, with exact answers for 90°, 180° and 270° and the working.

- Page: https://www.acalculator.org/math/rotation-calculator
- JSON spec: https://www.acalculator.org/math/rotation-calculator.json
- Version: 588e6a4ce2c8

## Default answer

Example with the default inputs (Angle (degrees) 90, Direction Counterclockwise, Centre x 0, Centre y 0, Points [x 3, y 2; x 5, y 2; x 4, y 6]): The image points are A′(−2, 3); B′(−2, 5); C′(−6, 4).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| angle | Angle (degrees) | How far to turn each point about the centre, in degrees. |
| dir | Direction | Counterclockwise (the usual positive direction) or clockwise. |
| cx | Centre x | The x coordinate of the centre of rotation; 0 when left empty. |
| cy | Centre y | The y coordinate of the centre of rotation; 0 when left empty. |
| points | Points | The points of the figure to rotate, A, B, C and so on, up to 12. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| image | Image points | Each point after the rotation, as A′(x, y); B′(x, y) and so on. |
| turn | Counterclockwise turn | The same rotation as a counterclockwise angle from 0° up to 360°. |
| rule | Rule about the centre | Where a point goes, measured from the centre: exact for a multiple of 90°. |
| steps | Working | For each point, x′ = cx + (x − cx)cos θ − (y − cy)sin θ and y′ = cy + (x − cx)sin θ + (y − cy)cos θ. |

## Method

x′ = cx + (x − cx)cos θ − (y − cy)sin θ, y′ = cy + (x − cx)sin θ + (y − cy)cos θ, with θ counterclockwise; clockwise is −θ.

## Assumptions

- The angle is reduced to a counterclockwise turn from 0° up to 360° before rotating.
- A multiple of 90° is exact: typed decimals are read exactly and each coordinate is rounded once to 10 significant figures.
- Any other angle uses double-precision cos and sin; a coordinate within 10⁻¹² of the figure’s size from 0 shows as 0.

## Worked examples

1. angle = 90, dir = ccw, points = {"x":3,"y":2} or undefined gives image = A′(−2, 3), rule = (x, y) → (−y, x), turn = 90. Source: CK-12 Foundation, Geometry, 8.11 Rotation Rules (counterclockwise about the origin: 90° (x, y) → (−y, x), 180° (−x, −y), 270° (y, −x); (3, 2) turned 90° is (−2, 3)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.11%3A_Rotation_Rules (retrieved 2026-10-05).
2. angle = 270, dir = ccw, points = {"x":-2,"y":6} or undefined gives image = A′(6, 2). Source: CK-12 Foundation, Geometry, 8.11 Rotation Rules (counterclockwise about the origin: 90° (x, y) → (−y, x), 180° (−x, −y), 270° (y, −x); (3, 2) turned 90° is (−2, 3)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.11%3A_Rotation_Rules (retrieved 2026-10-05): J(−2, 6) to J′(6, 2) is the 270° rule.
3. angle = 90, dir = cw, cx = 1, cy = 1, points = {"x":3,"y":2} or undefined gives image = A′(2, −1), turn = 270.
4. angle = 45, dir = ccw, points = {"x":2,"y":0} or undefined gives image = A′(1.414213562, 1.414213562), turn = 45. Source: OpenStax, Algebra and Trigonometry 2e, §12.4 Rotation of Axes (x = x′cos θ − y′sin θ, y = x′sin θ + y′cos θ), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-4-rotation-of-axes (retrieved 2026-10-05).
5. angle = -450, dir = ccw, points = {"x":0.1,"y":0.2} or undefined gives image = A′(0.2, −0.1), turn = 270.

## FAQ

### How do I rotate a point 90° counterclockwise about the origin?

Use (x, y) → (−y, x). The point (3, 2) turned 90° counterclockwise is (−2, 3).

### What are the 180° and 270° rotation rules?

A 180° turn sends (x, y) to (−x, −y). A 270° counterclockwise turn, the same as 90° clockwise, sends (x, y) to (y, −x): (−2, 6) goes to (6, 2).

### How do I rotate a point by any angle?

Use x′ = x cos θ − y sin θ and y′ = x sin θ + y cos θ for a counterclockwise angle θ about the origin. The point (2, 0) turned 45° goes to (2cos 45°, 2sin 45°) = (√2, √2) ≈ (1.4142, 1.4142).

### How do I rotate about a point that is not the origin?

Subtract the center, rotate, then add the center back: x′ = cx + (x − cx)cos θ − (y − cy)sin θ and y′ = cy + (x − cx)sin θ + (y − cy)cos θ. Turning (3, 2) 90° clockwise about (1, 1) gives (2, −1).

### Is a positive angle clockwise or counterclockwise?

Counterclockwise. In maths a positive angle turns from the positive x-axis toward the positive y-axis. A clockwise turn of θ is the same as a counterclockwise turn of −θ, or 360° − θ.

### Does a rotation change the size of a figure?

No. A rotation keeps every length, angle and area. Only the position and the direction the figure faces change.

## Sources

- CK-12 Foundation, Geometry, 8.11 Rotation Rules (counterclockwise rules for 90°, 180° and 270° about the origin; (3, 2) turned 90° is (−2, 3); J(−2, 6) to J′(6, 2) by 270°), on K12 LibreTexts. https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.11%3A_Rotation_Rules (retrieved 2026-10-05)
- OpenStax, Algebra and Trigonometry 2e, §12.4 Rotation of Axes (x = x′cos θ − y′sin θ, y = x′sin θ + y′cos θ), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-4-rotation-of-axes (retrieved 2026-10-05)