Rotation Calculator – Guide & Formulas
Rotate points around the origin or any center point. Calculate clockwise or counterclockwise rotation with angle in degrees or radians.
Rotate 2D points around the origin or any center point by any angle. Get the new coordinates for clockwise or counterclockwise rotation with step-by-step solutions.
Key Takeaway
Use the free Rotation Calculator to rotate points around the origin or any center point. calculate clockwise or counterclockwise rotation with angle in degrees or radians. Get instant results with step-by-step explanations.
How to Use the Rotation Calculator
- Enter the x and y coordinates of the point to rotate.
- Enter the center of rotation (default is the origin 0,0).
- Enter the rotation angle in degrees (positive for counterclockwise).
- Click "Calculate" to see the rotated coordinates.
The Formula
Variable Definitions
- x', y': Rotated coordinates of the point
- x, y: Original coordinates of the point
- h, k: Center of rotation coordinates
- θ: Rotation angle in degrees (positive = counterclockwise)
Example: Rotate (3, 4) by 90° Counterclockwise Around Origin
Rotate the point (3, 4) by 90° counterclockwise around the origin.
- Step 1: Identify values: x = 3, y = 4, h = 0, k = 0, θ = 90°.
- Step 2: cos(90°) = 0, sin(90°) = 1.
- Step 3: x' = (3-0)(0) - (4-0)(1) + 0 = -4.
- Step 4: y' = (3-0)(1) + (4-0)(0) + 0 = 3.
- Step 5: Rotated point is (-4, 3).
Frequently Asked Questions
What is a 2D rotation?
A 2D rotation transforms a point around a center by a specified angle. The point maintains its distance from the center while changing its angular position.
What is the rotation formula?
The formula is x' = (x-h)cos(θ) - (y-k)sin(θ) + h and y' = (x-h)sin(θ) + (y-k)cos(θ) + k, where (h,k) is the center and θ is the angle.
What does positive rotation mean?
Positive rotation is counterclockwise by convention. Negative rotation is clockwise.
How do I rotate 90 degrees?
For 90° counterclockwise around the origin: (x,y) becomes (-y,x). For 90° clockwise: (x,y) becomes (y,-x).
What is a rotation matrix?
A rotation matrix is a 2x2 matrix [[cos θ, -sin θ], [sin θ, cos θ]] that multiplies the point vector to produce the rotated coordinates.
How do I rotate around a point that is not the origin?
Translate the point so the center of rotation is at the origin, rotate, then translate back. This is what the formula handles automatically.
Can I rotate by any angle?
Yes, you can rotate by any angle in degrees or radians. Common angles are 90°, 180°, 270°, and 360°.
What happens when I rotate 360 degrees?
A 360° rotation returns the point to its original position. The coordinates remain unchanged.
How is rotation used in computer graphics?
Rotation is fundamental in computer graphics for animating objects, camera movements, and transforming sprites in 2D games.
What is the difference between rotation and reflection?
Rotation spins a point around a center, preserving distance. Reflection flips a point across a line, creating a mirror image.