90 Degrees Counterclockwise Rotation Calculator

This calculator helps you determine the new coordinates of a point after rotating it 90 degrees counterclockwise around another reference point. This is commonly used in computer graphics, physics simulations, and geometric transformations.

How to Use This Calculator

To calculate the rotated coordinates:

Enter the original coordinates of the point you want to rotate (X₁, Y₁).
Enter the coordinates of the center point (X₀, Y₀) around which the rotation will occur.
Click "Calculate" to see the new coordinates after rotation.
The result will show the new X and Y coordinates of the rotated point.

The calculator will also display a visualization of the rotation for better understanding.

Formula

To rotate a point (X₁, Y₁) 90 degrees counterclockwise around another point (X₀, Y₀), use the following transformation:

X' = X₀ - (Y₁ - Y₀) Y' = Y₀ + (X₁ - X₀)

Where:

(X₁, Y₁) are the original coordinates of the point to be rotated
(X₀, Y₀) are the coordinates of the center point
(X', Y') are the new coordinates after rotation

This formula effectively translates the point relative to the center, performs the rotation, and then translates back.

Worked Example

Let's say we want to rotate the point (4, 6) 90 degrees counterclockwise around the point (2, 3).

First, translate the point relative to the center:
X_rel = X₁ - X₀ = 4 - 2 = 2 Y_rel = Y₁ - Y₀ = 6 - 3 = 3
Apply the 90-degree counterclockwise rotation:
X_rot = -Y_rel = -3 Y_rot = X_rel = 2
Translate back to the original coordinate system:
X' = X₀ + X_rot = 2 + (-3) = -1 Y' = Y₀ + Y_rot = 3 + 2 = 5

The rotated point is (-1, 5).

FAQ

What does a 90-degree counterclockwise rotation do?: A 90-degree counterclockwise rotation moves every point to the left relative to the center of rotation, effectively swapping the x and y coordinates and changing their signs.
Can I rotate a point clockwise with this calculator?: No, this calculator specifically handles 90-degree counterclockwise rotations. For clockwise rotations, you would need a different calculator or formula.
What if I want to rotate multiple points?: You can use this calculator for each point individually. For rotating multiple points around the same center, you can simply apply the same formula to each point's coordinates.
Is this formula the same for all coordinate systems?: Yes, the formula works the same way in any Cartesian coordinate system, whether it's standard (x right, y up) or any other orientation.