Where do points go in a rotation?
Type the angle, the direction, the center of rotation and the points of your figure. The rotation calculator turns each point about the center and shows the rule and the working, exactly for multiples of 90°.
- Image points
- A′(−2, 3); B′(−2, 5); C′(−6, 4)
The image points are A′(−2, 3); B′(−2, 5); C′(−6, 4).
- Counterclockwise turn
- 90
- Rule about the centre
- (x, y) → (−y, x)
- Working
- centre (0, 0), counterclockwise turn 90°; (x, y) → (−y, x) about the centre; A(3, 2) → A′(−2, 3); B(5, 2) → B′(−2, 5); C(4, 6) → C′(−6, 4)
Image points: A′(−2, 3); B′(−2, 5); C′(−6, 4). The image points are A′(−2, 3); B′(−2, 5); C′(−6, 4).
How it is worked out
How to calculate
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.
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).
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.
- 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
Each example is checked against the calculator on every build.
- Angle (degrees) 90, Direction Counterclockwise, Points 3 2 gives Image points A′(−2, 3), Rule about the centre (x, y) → (−y, x), Counterclockwise 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)
- Angle (degrees) 270, Direction Counterclockwise, Points -2 6 gives Image points 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
- Angle (degrees) 90, Direction Clockwise, Centre x 1, Centre y 1, Points 3 2 gives Image points A′(2, −1), Counterclockwise turn 270.
- Angle (degrees) 45, Direction Counterclockwise, Points 2 0 gives Image points A′(1.414213562, 1.414213562), Counterclockwise 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)
- Angle (degrees) -450, Direction Counterclockwise, Points 0.1 0.2 gives Image points A′(0.2, −0.1), Counterclockwise turn 270.
How it works
A rotation by θ counterclockwise about the center C = (cx, cy) sends each point (x, y) to
- x′ = cx + (x − cx)cos θ − (y − cy)sin θ
- y′ = cy + (x − cx)sin θ + (y − cy)cos θ
A clockwise angle is turned into the counterclockwise angle −θ. The page then reduces the angle to a counterclockwise turn from 0° up to 360° (−450° becomes 270°), exactly from the typed decimal, and shows it as the counterclockwise turn.
Multiples of 90°. cos θ and sin θ are 0, 1 or −1, so the image is exact:
| Counterclockwise turn | Image of (x, y) about the origin |
|---|---|
| 0° | (x, y) |
| 90° | (−y, x) |
| 180° | (−x, −y) |
| 270° | (y, −x) |
About another center, the same rule applies to (x − cx, y − cy), and the center is added back.
Rules
- The angle is from −10⁹ to 10⁹ degrees. Every coordinate is from −10¹² to 10¹².
- The center is (0, 0) when its boxes are left empty.
- Up to 12 points, named A, B, C and so on.
Output format. For a multiple of 90°, typed decimals are read exactly and each coordinate is rounded once to 10 significant figures. For any other angle, the turn is converted to radians as turn × π ÷ 180 and the coordinates use double-precision cos and sin; a coordinate whose size is at most 10⁻¹² times the largest of |x − cx|, |y − cy|, |cx| and |cy| for that point shows as 0. Negative values use the true minus sign (−).
Worked examples by hand
90° counterclockwise about the origin. (3, 2) → (−2, 3).
270° counterclockwise. (−2, 6) → (6, −(−2)) = (6, 2).
90° clockwise about (1, 1). This is a 270° counterclockwise turn. (3, 2) is (2, 1) from the center; (x, y) → (y, −x) gives (1, −2); adding the center gives (2, −1).
45° about the origin. (2, 0) → (2cos 45°, 2sin 45°) = (√2, √2) = (1.414213562, 1.414213562).
−450° counterclockwise. −450° + 2 × 360° = 270°, so (0.1, 0.2) → (0.2, −0.1).
Other questions people ask
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.