Rotation Calculator
Calculate rotations of points around any center of rotation by any angle with step-by-step solutions and visual diagrams.
Understanding Rotations
What is a Rotation?
A rotation is a transformation that turns a point or shape around a fixed point (center of rotation) by a specified angle. Positive angles indicate counterclockwise rotation, while negative angles indicate clockwise rotation.
Rotation Formulas
General Formula (about origin):
x' = x cos θ - y sin θ
y' = x sin θ + y cos θ
Common Rotations (counterclockwise about origin):
90°: (x, y) → (-y, x)
180°: (x, y) → (-x, -y)
270°: (x, y) → (y, -x)
-90° (clockwise): (x, y) → (y, -x)
Properties of Rotations
- Preserved: Distance (isometry), angle measures, shape, size, and orientation
- Changed: Position and direction
- Every point moves along a circular arc centered at the center of rotation
- The center of rotation is the only point that doesn't move
- Rotating 360° returns the shape to its original position
Real-World Applications
- Mechanics: Wheels, gears, and rotating machinery
- Astronomy: Earth's rotation, planetary orbits
- Computer Graphics: Rotating objects in games and animations
- Engineering: Analyzing forces on rotating parts
- Navigation: Compass bearings and course changes
- Architecture: Circular or spiral designs
Frequently Asked Questions
How do you rotate a point 90 degrees counterclockwise?
For rotation about the origin, use the formula (x, y) → (-y, x). For example, (3, 2) rotated 90° becomes (-2, 3). If rotating about a different center, first translate to the origin, rotate, then translate back.
What's the difference between clockwise and counterclockwise rotation?
Counterclockwise (positive angles) rotates in the direction opposite to clock hands. Clockwise (negative angles) rotates in the same direction as clock hands. A 90° counterclockwise rotation equals a -270° clockwise rotation.
How do you rotate about a point other than the origin?
Use a three-step process: (1) Translate the shape so the center of rotation moves to the origin, (2) Rotate using the standard formulas, (3) Translate back by the same amount in the opposite direction.
Is a 180° rotation the same as reflection?
A 180° rotation about the origin gives the same result as point reflection through the origin: (x, y) → (-x, -y). However, the transformation is conceptually different—rotation preserves orientation while reflection reverses it.
Why do we use radians in rotation formulas?
Trigonometric functions (sin, cos) in calculators and programming use radians by default. To convert degrees to radians, multiply by π/180. For example, 90° = 90 × π/180 = π/2 radians ≈ 1.5708 radians.
Can you combine multiple rotations?
Yes! Multiple rotations about the same center can be combined by adding the angles. For example, rotating 45° then 30° equals a single 75° rotation. However, rotations about different centers don't simply add.
What shapes look the same after rotation?
Shapes with rotational symmetry appear unchanged after certain rotations. A square looks the same after 90°, 180°, and 270° rotations. A circle looks the same after any rotation. Regular polygons have rotational symmetry.
How do rotations preserve distance?
Rotations are rigid transformations (isometries) that preserve all distances and angles. The distance between any two points remains the same after rotation, making rotation useful in geometry, physics, and engineering.