# Rotation Calculator

URL: https://speedycalculator.com/calculators/math/rotation/
Last updated: 2026-10-30

> The rotation calculator rotates up to four points by an angle θ about a pivot point (the origin by default) and gives their new coordinates.

## Inputs

- Angle of rotation θ (counter-clockwise) (°, rad, π rad, grad, turn)
- Pivot – x₀
- Pivot – y₀
- Number of points
- Point 1 – x
- Point 1 – y
- Point 2 – x
- Point 2 – y
- Point 3 – x
- Point 3 – y
- Point 4 – x
- Point 4 – y

## Results

- Point 1 – x'
- Point 1 – y'
- Point 2 – x'
- Point 2 – y'
- Point 3 – x'
- Point 3 – y'
- Point 4 – x'
- Point 4 – y'

## Formula

```
x′ = x₀ + (x − x₀)cos θ − (y − y₀)sin θ
y′ = y₀ + (x − x₀)sin θ + (y − y₀)cos θ
```

## Explanation

Rotations are used in geometry transformations, computer graphics, robotics and surveying. A positive angle turns counter-clockwise; a negative angle turns clockwise.

### How to use the Rotation Calculator

- **Angle of rotation θ (counter-clockwise)**
- **Pivot – x₀**
- **Pivot – y₀**
- **Number of points**
- **Point 1 – x**
- **Point 1 – y**
- **Point 2 – x**
- **Point 2 – y**
- **Point 3 – x**
- **Point 3 – y**
- **Point 4 – x**
- **Point 4 – y**

Results update as you type, and everything is calculated in your browser.

### How it works

Each point is moved so the pivot is at the origin, rotated with the rotation matrix and moved back.

### Inputs explained

FieldWhat to enterAngle of rotation θ (counter-clockwise)Units: °, rad, π rad, grad, turnPivot – x₀A numberPivot – y₀A numberNumber of pointsChoose one: 1, 2, 3, 4Point 1 – xA numberPoint 1 – yA numberPoint 2 – xA numberPoint 2 – yA numberPoint 3 – xA numberPoint 3 – yA numberPoint 4 – xA numberPoint 4 – yA number

### What each result means

- **Point 1 – x'** = [Pivot – x₀] + ([Point 1 – x] - [Pivot – x₀])×cos(rad([Angle of rotation θ (counter-clockwise)])) - ([Point 1 – y] - [Pivot – y₀])×sin(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 1 – y'** = [Pivot – y₀] + ([Point 1 – x] - [Pivot – x₀])×sin(rad([Angle of rotation θ (counter-clockwise)])) + ([Point 1 – y] - [Pivot – y₀])×cos(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 2 – x'** = [Pivot – x₀] + ([Point 2 – x] - [Pivot – x₀])×cos(rad([Angle of rotation θ (counter-clockwise)])) - ([Point 2 – y] - [Pivot – y₀])×sin(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 2 – y'** = [Pivot – y₀] + ([Point 2 – x] - [Pivot – x₀])×sin(rad([Angle of rotation θ (counter-clockwise)])) + ([Point 2 – y] - [Pivot – y₀])×cos(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 3 – x'** = [Pivot – x₀] + ([Point 3 – x] - [Pivot – x₀])×cos(rad([Angle of rotation θ (counter-clockwise)])) - ([Point 3 – y] - [Pivot – y₀])×sin(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 3 – y'** = [Pivot – y₀] + ([Point 3 – x] - [Pivot – x₀])×sin(rad([Angle of rotation θ (counter-clockwise)])) + ([Point 3 – y] - [Pivot – y₀])×cos(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 4 – x'** = [Pivot – x₀] + ([Point 4 – x] - [Pivot – x₀])×cos(rad([Angle of rotation θ (counter-clockwise)])) - ([Point 4 – y] - [Pivot – y₀])×sin(rad([Angle of rotation θ (counter-clockwise)]))
- **Point 4 – y'** = [Pivot – y₀] + ([Point 4 – x] - [Pivot – x₀])×sin(rad([Angle of rotation θ (counter-clockwise)])) + ([Point 4 – y] - [Pivot – y₀])×cos(rad([Angle of rotation θ (counter-clockwise)]))

### Example values

How the main result changes when **Angle of rotation θ** changes and everything else stays at the example values above:

Angle of rotation θ (counter-clockwise) (°)Point 1 – x'451.414267.50.224290-1112.5-2.0719135-2.8284180-3

## Worked example

Inputs — Angle of rotation θ (counter-clockwise): 90 °; Pivot – x₀: 0; Pivot – y₀: 0; Number of points: 2; Point 1 – x: 3; Point 1 – y: 1; Point 2 – x: 5; Point 2 – y: 4.

Results — **Point 1 – x' = -1**; Point 1 – y' = 3; Point 2 – x' = -4; Point 2 – y' = 5.

## Key terms

- **Rotation matrix**: [[cos θ, −sin θ], [sin θ, cos θ]].

## FAQ

**Which way is a positive rotation?**
Counter-clockwise.

**What does a 90° rotation do to (x, y)?**
It maps it to (−y, x) about the origin.

**What about 180°?**
(x, y) → (−x, −y).

**Does rotation change distances?**
No – rotations preserve lengths and angles.


## Sources

- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Rotation matrix](https://mathworld.wolfram.com/RotationMatrix.html)