🔄 Rotation Calculator
Rotate points around a center point
Point to Rotate (x, y)
Rotation Center (default: origin)
Positive = counterclockwise, Negative = clockwise
How to Use This Calculator
Enter Point to Rotate
Input the coordinates (x, y) of the point you want to rotate.
Enter Rotation Center (Optional)
Input the coordinates of the rotation center (default is origin 0, 0).
Enter Rotation Angle
Input the rotation angle in degrees. Positive values rotate counterclockwise; negative values rotate clockwise.
Calculate
Press "Calculate Rotation" to find the coordinates of the rotated point.
Formula
Rotation Matrix:
[x'] = [cos θ -sin θ] [x - cx] + [cx] [y'] [sin θ cos θ] [y - cy] [cy]
where (cx, cy) is the rotation center and θ is the angle
Steps:
- Translate point so center is at origin: (x - cx, y - cy)
- Rotate around origin: x' = (x-cx)cos θ - (y-cy)sin θ, y' = (x-cx)sin θ + (y-cy)cos θ
- Translate back: (x' + cx, y' + cy)
Example: Rotate (1, 0) 90° around origin
x' = 1×cos(90°) - 0×sin(90°) = 0
y' = 1×sin(90°) + 0×cos(90°) = 1
Result: (0, 1)
About Rotation Calculator
The Rotation Calculator rotates a point around a center point by a specified angle. It uses rotation matrix transformations to calculate the new coordinates after rotation.
When to Use This Calculator
- Geometry: Rotate points and shapes in coordinate geometry
- Computer Graphics: Transform objects in 2D rendering
- Engineering: Calculate rotated positions in designs
- Mathematics: Study rotation transformations
- Game Development: Rotate sprites and game objects
Why Use Our Calculator?
- ✅ Flexible Center: Rotate around any point, not just origin
- ✅ Precise Calculations: Uses rotation matrix for accuracy
- ✅ Direction Support: Handles both clockwise and counterclockwise rotations
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Rotation
Rotation is a transformation that turns a point around a center point by a given angle. In 2D, rotation preserves distances and angles but changes positions.
- Positive angle: Counterclockwise rotation
- Negative angle: Clockwise rotation
- 90° rotation: (x, y) → (-y, x) for origin-centered
- 180° rotation: (x, y) → (-x, -y) for origin-centered
- Rotation preserves distance from center
Frequently Asked Questions
How do I rotate a point around the origin?
Set the center coordinates to (0, 0). The rotation formulas become: x' = x cos θ - y sin θ, y' = x sin θ + y cos θ.
What's the difference between clockwise and counterclockwise rotation?
Counterclockwise rotation uses positive angles. Clockwise rotation uses negative angles. For example, 90° counterclockwise vs -90° clockwise.
How do I rotate around a point that's not the origin?
Use the three-step process: (1) Translate so center is at origin, (2) Rotate, (3) Translate back. This calculator does all steps automatically.
What happens when I rotate 360°?
A 360° rotation returns the point to its original position. Any multiple of 360° (720°, 1080°, etc.) also returns to the original position.
Can I rotate multiple points at once?
This calculator rotates one point at a time. To rotate multiple points, calculate each point separately using the same center and angle.