The Shape Rotation Calculator calculates the new coordinates of a shape after it is turned by an angle around a center point.
Each point is moved using x′ = cx + (x − cx)cos θ − (y − cy)sin θ and y′ = cy + (x − cx)sin θ + (y − cy)cos θ. Enter the points, the angle, the direction (counterclockwise or clockwise) and the center — the origin (0, 0) or any other point — and the calculator lists the image points A′, B′, C′… and draws both shapes.
Shape Rotation Calculator
Rotate a shape’s points by any angle about the origin or any centre point, clockwise or anticlockwise, and see the image on a graph.
Formula
x′ = cx + (x − cx) cos θ − (y − cy) sin θ
y′ = cy + (x − cx) sin θ + (y − cy) cos θ
Where:
- (x, y) = original point; (x′, y′) = rotated point
- (cx, cy) = center of rotation
- θ = rotation angle; positive = counterclockwise, negative = clockwise
How the Calculation Works
Step 1: Shift each point so the center of rotation is at the origin: (x − cx, y − cy).
Step 2: Rotate it with the cos/sin formula.
Step 3: Shift it back by adding (cx, cy). Distances and angles are preserved, so the shape’s size doesn’t change.
Rotation Rules About the Origin
| Rotation | Rule |
|---|---|
| 90° counterclockwise (= 270° clockwise) | (x, y) → (−y, x) |
| 180° (either direction) | (x, y) → (−x, −y) |
| 270° counterclockwise (= 90° clockwise) | (x, y) → (y, −x) |
| 360° | (x, y) → (x, y) |
Variables Explained
| Input | Meaning |
|---|---|
| Points | One x, y pair per line, e.g. 2, 3 |
| Angle | Amount of turn, in degrees |
| Direction | Counterclockwise (anticlockwise) or clockwise |
| Center x, y | Point the shape turns around (0, 0 for the origin) |
Example Calculation
90° counterclockwise about the origin
Given: A(1, 1), B(5, 1), C(2, 4)
Rule: (x, y) → (−y, x)
Result: A′(−1, 1), B′(−1, 5), C′(−4, 2)
90° counterclockwise about (2, 3)
Given: B(5, 1)
Shift: (5 − 2, 1 − 3) = (3, −2); rotate: (2, 3); shift back: (2 + 2, 3 + 3)
Result: B′(4, 6)
Shape Properties
A rotation is a rigid transformation: the image is congruent to the original, with the same side lengths, angles and area. Only its position and orientation change. The center of rotation is the one point that does not move.
Dimensions Explained
Clockwise vs counterclockwise
In math, a positive angle means counterclockwise. A 90° clockwise turn is the same as a 270° counterclockwise turn.
Decimal results
Angles such as 45° or 30° give decimal coordinates because they involve √2 or √3. The calculator rounds tiny floating-point errors so 90° turns give clean whole numbers.
Frequently Asked Questions
What is the rule for a 90° counterclockwise rotation?
(x, y) → (−y, x) about the origin.
What is the rule for a 90° clockwise rotation?
(x, y) → (y, −x) about the origin.
What is the rule for a 180° rotation?
(x, y) → (−x, −y). Direction doesn’t matter for 180°.
How do you rotate a shape around a point that is not the origin?
Subtract the center, rotate, then add the center back — or enter the center in the calculator.
Does rotation change the size of a shape?
No. Rotations preserve lengths, angles and area.