Rotation Calculator
Free online rotation calculator -- rotate any 2D point by any angle around the origin or a custom pivot. Get instant rotated coordinates with an interactive chart and rotation matrix.
Pivot point (optional, defaults to origin)
About This Calculator
The Rotation Calculator is a geometry tool that rotates any 2D point around the origin or a custom pivot point by any angle. Enter the X and Y coordinates of a point, choose an angle in degrees, and instantly see the rotated coordinates on an interactive scatter plot. You can also set a custom pivot point for rotation around any location in the plane.
Rotation is a fundamental transformation in coordinate geometry and computer graphics. The calculator uses the standard rotation formula: x' = x cos(theta) - y sin(theta) and y' = x sin(theta) + y cos(theta) for rotation around the origin. For a custom pivot point (px, py), the calculator first translates the point relative to the pivot, applies the rotation, and then translates back. Positive angles produce counterclockwise rotations, while negative angles produce clockwise rotations. The rotation matrix is displayed alongside results for educational reference.
Common use cases include rotating shapes in geometry problems, transforming coordinates in computer graphics and game development, calculating rotated positions in engineering (robotics arm endpoints, rotated sensor mounts), and solving physics problems involving angular displacement. Students and teachers can verify rotation rules manually and compare with the matrix output for better understanding of linear transformations.
Regional Notes
India: Rotation concepts are covered in CBSE and ICSE Class 10-12 coordinate geometry and linear algebra curricula. The rotation matrix approach is introduced in Class 12 mathematics.
United States: Rotation transformations are taught in high school geometry (CCSS HSG-CO.A.2-5) and are fundamental to computer graphics courses at the college level.
United Kingdom: Rotation is part of the GCSE and A-Level mathematics curriculum. Students learn to describe rotations with center, angle, and direction, and use matrices for rotations at A-Level Further Mathematics.
Frequently Asked Questions
How do you rotate a point 90 degrees clockwise?
To rotate a point (x, y) 90 degrees clockwise, use the formula x' = y and y' = -x. Alternatively, enter -90 degrees in the Rotation Calculator to get the exact coordinates of any point rotated 90 degrees clockwise.
What is the formula for rotating a point around the origin?
The formula for rotating a point (x, y) by angle theta around the origin is: x' = x cos(theta) - y sin(theta) and y' = x sin(theta) + y cos(theta). Positive angles rotate counterclockwise, negative angles rotate clockwise.
How do you rotate a point around another point (pivot)?
To rotate a point (x, y) around a pivot (px, py), first translate the point by subtracting the pivot: dx = x - px, dy = y - py. Then apply the rotation: dx' = dx cos(theta) - dy sin(theta), dy' = dx sin(theta) + dy cos(theta). Finally translate back: x' = px + dx', y' = py + dy'.
What is the rotation matrix?
The rotation matrix for a 2D rotation by angle theta is R = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]. When you multiply this matrix by the vector (x, y), you get the rotated coordinates (x', y'). The Rotation Calculator displays this matrix for every calculation.
What is the difference between clockwise and counterclockwise rotation?
Counterclockwise rotation uses positive angles and follows the formula x' = x cos(theta) - y sin(theta), y' = x sin(theta) + y cos(theta). Clockwise rotation uses negative angles, equivalent to rotating by -theta. The Rotation Calculator accepts both: positive for counterclockwise, negative for clockwise.
What are common rotation angles used in geometry?
Common rotation angles are 90 deg (right angle), 180 deg (half turn), 270 deg (three-quarter turn), and 360 deg (full turn). A 90 deg counterclockwise rotation transforms (x, y) to (-y, x). A 180 deg rotation transforms (x, y) to (-x, -y). A 270 deg rotation transforms (x, y) to (y, -x).
Is rotation an isometric transformation?
Yes, rotation is an isometric transformation (also called a rigid motion or congruence transformation). This means distances between points are preserved after rotation -- the shape and size of any object remain unchanged. Rotations also preserve angles and orientation (handedness).
What is the rotation rule for 180 degrees?
The rotation rule for 180 degrees is (x, y) -> (-x, -y). Both coordinates are negated. This means a point (3, 4) rotated 180 deg becomes (-3, -4). A 180 deg rotation is the same whether you rotate clockwise or counterclockwise.