What it does
This calculator rotates a point (x, y) by any angle, clockwise or counterclockwise, around the origin or around any center point you choose. It shows the new coordinates, the rotation matrix with its numbers filled in, and each step. Use it for geometry homework, transformations, game and graphics code, or CAD work.
How it's calculated
A counterclockwise rotation by θ about the origin multiplies the point by the rotation matrix:
[y′] = [sin θ cos θ] [y]
A clockwise rotation is the same with −θ. To rotate around a center (h, k), shift the point so the center is at the origin, rotate, then shift back:
Worked example
Rotate (5, 2) by 90° clockwise around (1, 1). Shift: (5 − 1, 2 − 1) = (4, 1). Clockwise 90° means θ = −90°, so cos θ = 0 and sin θ = −1. Then x = 4·0 − 1·(−1) = 1 and y = 4·(−1) + 1·0 = −4. Shift back: (1 + 1, −4 + 1) = (2, −3). The distance from the center stays √17 ≈ 4.123 before and after.
Rotating a whole shape
To rotate a triangle, rectangle or any polygon, rotate each corner (vertex) with the same angle and center, then join the new corners in the same order. The shape keeps its size and angles; only its position and direction change. Rotating about one of the shape's own corners keeps that corner fixed, which is how you'd turn a door or a hand on a clock face.
Quick rules for common angles
| Rotation about the origin | (x, y) becomes |
|---|---|
| 90° counterclockwise (270° clockwise) | (−y, x) |
| 180° | (−x, −y) |
| 90° clockwise (270° counterclockwise) | (y, −x) |
| 360° | (x, y) |
Limitations
- This rotates a point in two dimensions. 3D rotations need a 3 × 3 matrix and an axis.
- It moves the point and keeps the axes fixed. Rotating the axes instead (a change of coordinates) uses the transposed matrix, which is the same as rotating the point the other way.
- Answers are rounded to 10 significant digits. Multiples of 90° in degrees are exact.