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Rotation — Definition, Formula & Examples

Rotation

A transformation in which a plane figure turns around a fixed center point. In other words, one point on the plane, the center of rotation, is fixed and everything else on the plane rotates about that point by a given angle.

A 5-vertex pre-image arrow on the left rotated about a fixed centre point to form the image on the right; dashed terracotta arc-arrows show the rotation.

See also

Pre-image, image

Key Formula

x′=(x−a)cos⁡θ−(y−b)sin⁡θ+ay′=(x−a)sin⁡θ+(y−b)cos⁡θ+b\begin{aligned} x' &= (x - a)\cos\theta - (y - b)\sin\theta + a \\ y' &= (x - a)\sin\theta + (y - b)\cos\theta + b \end{aligned}
Where:
  • (x,y)(x, y) = Coordinates of the original point (pre-image)
  • (x′,y′)(x', y') = Coordinates of the rotated point (image)
  • (a,b)(a, b) = Coordinates of the center of rotation
  • θ\theta = Angle of rotation (positive = counterclockwise, negative = clockwise)

Worked Example

Problem: Rotate the point P(3, 1) by 90° counterclockwise about the origin.
Step 1: Identify the center of rotation and angle. The center is the origin (0, 0) and the angle is θ = 90°.
(a,b)=(0,0),θ=90°(a, b) = (0, 0), \quad \theta = 90°
Step 2: Since the center is the origin, the general formulas simplify. Recall that cos 90° = 0 and sin 90° = 1.
x′=xcos⁡90°−ysin⁡90°=x(0)−y(1)=−yx' = x\cos 90° - y\sin 90° = x(0) - y(1) = -y
Step 3: Compute the new y-coordinate using the same trigonometric values.
y′=xsin⁡90°+ycos⁡90°=x(1)+y(0)=xy' = x\sin 90° + y\cos 90° = x(1) + y(0) = x
Step 4: Substitute x = 3 and y = 1 into the simplified formulas.
x′=−1,y′=3x' = -1, \quad y' = 3
Answer: The image of P(3, 1) after a 90° counterclockwise rotation about the origin is P'(−1, 3).

Another Example

This example shows rotation about a center other than the origin, requiring a translate-rotate-translate back approach.

Problem: Rotate the point A(5, 2) by 90° counterclockwise about the center C(1, 1).
Step 1: Identify the values. The center of rotation is (a, b) = (1, 1), the angle is θ = 90°, and the point is (x, y) = (5, 2).
Step 2: Translate the point so the center moves to the origin by subtracting (a, b).
(x−a,  y−b)=(5−1,  2−1)=(4,1)(x - a,\; y - b) = (5 - 1,\; 2 - 1) = (4, 1)
Step 3: Apply the 90° counterclockwise rotation rule (x, y) → (−y, x) to the translated point.
(4,1)→(−1,4)(4, 1) \to (-1, 4)
Step 4: Translate back by adding (a, b) to get the final image point.
x′=−1+1=0,y′=4+1=5x' = -1 + 1 = 0, \quad y' = 4 + 1 = 5
Answer: The image of A(5, 2) after a 90° counterclockwise rotation about (1, 1) is A'(0, 5).

Frequently Asked Questions

What is the rule for 90°, 180°, and 270° rotations about the origin?
For counterclockwise rotations about the origin: a 90° rotation maps (x, y) to (−y, x); a 180° rotation maps (x, y) to (−x, −y); a 270° rotation maps (x, y) to (y, −x). These shortcuts avoid using trigonometry and are the most commonly tested rules on exams.
Is rotation clockwise or counterclockwise?
By convention, a positive angle of rotation means counterclockwise, and a negative angle means clockwise. For example, a rotation of −90° is the same as a 90° clockwise rotation. Always check whether your problem specifies direction.
Does rotation change the size or shape of a figure?
No. Rotation is a rigid transformation (also called an isometry), which means it preserves distances and angle measures. The image is congruent to the pre-image — same size, same shape, just turned to a new orientation.

Rotation vs. Reflection

RotationReflection
DefinitionTurns a figure around a center point by an angleFlips a figure across a line of reflection
What stays the sameShape, size, and distance from centerShape, size, and distance from the line
OrientationPreserved (same handedness)Reversed (opposite handedness)
Key parametersCenter point and angleLine of reflection
TypeRigid transformation (isometry)Rigid transformation (isometry)

Why It Matters

Rotation appears throughout geometry courses when studying congruence, symmetry, and coordinate transformations. Many standardized tests (SAT, ACT, state exams) ask you to find the image of a point or polygon after a rotation about the origin. Beyond the classroom, rotations are fundamental in computer graphics, engineering design, and physics — anytime an object spins or changes orientation.

Common Mistakes

Mistake: Confusing clockwise and counterclockwise direction, especially mixing up a 90° CW rotation with a 90° CCW rotation.
Correction: Remember that positive angles rotate counterclockwise and negative angles rotate clockwise. A 90° CCW rotation sends (x, y) to (−y, x), while a 90° CW rotation sends (x, y) to (y, −x). These give different images, so always check the direction stated in the problem.
Mistake: Applying the origin rotation rules (x, y) → (−y, x) when the center of rotation is not the origin.
Correction: If the center is a point (a, b) other than (0, 0), you must first translate the point by subtracting (a, b), then apply the rotation, then translate back by adding (a, b). Skipping the translation steps produces the wrong answer.

Related Terms