Mathwords: Center of Rotation

Key Formula

\begin{cases} x' = (x - h)\cos\theta - (y - k)\sin\theta + h \\ y' = (x - h)\sin\theta + (y - k)\cos\theta + k \end{cases}

Where:

$(h, k)$ = The center of rotation — the fixed point that does not move
$(x, y)$ = The coordinates of the original point before the rotation
$(x', y')$ = The coordinates of the image point after the rotation
$\theta$ = The angle of rotation, measured counterclockwise from the positive x-axis

Worked Example

Problem: Rotate the point A(4, 1) by 90° counterclockwise about the center of rotation C(1, 1). Find the image A'.

Step 1: Identify the center of rotation and the angle. Here the center is C(1, 1) and the angle is θ = 90°.

(h, k) = (1, 1), \quad \theta = 90°

Step 2: Translate the point relative to the center by subtracting (h, k) from (x, y).

(x - h,\; y - k) = (4 - 1,\; 1 - 1) = (3, 0)

Step 3: Apply the rotation formulas using cos 90° = 0 and sin 90° = 1.

\begin{gathered}x' - h = 3\cos 90° - 0\sin 90° = 3(0) - 0(1) = 0 \\ y' - k = 3\sin 90° + 0\cos 90° = 3(1) + 0(0) = 3\end{gathered}

Step 4: Translate back by adding the center coordinates to get the final image.

(x', y') = (0 + 1,\; 3 + 1) = (1, 4)

Step 5: Verify: the distance from C to A is 3, and the distance from C to A' is also 3. The angle from CA to CA' is 90° counterclockwise. ✓

CA = \sqrt{(4-1)^2 + (1-1)^2} = 3, \quad CA' = \sqrt{(1-1)^2 + (4-1)^2} = 3

Answer: The image of A(4, 1) after a 90° counterclockwise rotation about C(1, 1) is A'(1, 4).

Another Example

This example works in reverse — instead of computing the image, you verify a proposed center of rotation and determine the angle. This is the type of problem where you need to find or confirm the center.

Problem: Two points are given: P(2, 0) and its image P'(0, 2) after a rotation. Verify that the origin (0, 0) is the center of rotation and find the angle.

Step 1: For a point to be the center of rotation, it must be equidistant from the original point and its image.

OP = \sqrt{2^2 + 0^2} = 2, \quad OP' = \sqrt{0^2 + 2^2} = 2

Step 2: The distances are equal, so the origin could be the center. Now find the angle from P to P' about the origin.

\text{Angle of } P = \arctan\!\left(\frac{0}{2}\right) = 0°, \quad \text{Angle of } P' = \arctan\!\left(\frac{2}{0}\right) = 90°

Step 3: The difference in angles gives the rotation angle.

\theta = 90° - 0° = 90° \text{ counterclockwise}

Step 4: Confirm with the rotation formula at the origin (h = 0, k = 0).

x' = 2\cos 90° - 0\sin 90° = 0, \quad y' = 2\sin 90° + 0\cos 90° = 2 \quad \checkmark

Answer: The origin (0, 0) is confirmed as the center of rotation, and the angle is 90° counterclockwise.

Frequently Asked Questions

How do you find the center of rotation given a point and its image?

The center of rotation lies on the perpendicular bisector of the segment connecting each original point to its image. If you have two such point-image pairs, find the perpendicular bisector of each segment. The center is where the two perpendicular bisectors intersect. This works because the center is equidistant from a point and its image.

Can the center of rotation be outside the figure?

Yes. The center of rotation can be inside, on, or completely outside the figure being rotated. When you rotate a shape about one of its own vertices, that vertex is the center. When you rotate a shape about the origin, the center may be far from the figure. The only requirement is that every point stays the same distance from the center after the rotation.

What is the difference between center of rotation and angle of rotation?

The center of rotation is the fixed point the figure turns around — it specifies where. The angle of rotation tells you how far the figure turns — it specifies how much. Both pieces of information are needed to fully describe a rotation. Without the center, you know how far to turn but not around what point; without the angle, you know the pivot but not the amount of turning.

Center of Rotation vs. Center of Dilation

	Center of Rotation	Center of Dilation
Definition	The fixed point about which a figure rotates	The fixed point from which a figure is scaled (enlarged or reduced)
Transformation type	Rotation (rigid motion — preserves size and shape)	Dilation (non-rigid — preserves shape but changes size)
What stays the same	Distance from center to any point is preserved	Angles are preserved; distances are multiplied by the scale factor
Key parameter	Angle of rotation θ	Scale factor k
Path of each point	Circular arc centered at the fixed point	Straight ray from or toward the fixed point

Why It Matters

You encounter the center of rotation in coordinate geometry, congruence proofs, and standardized tests whenever a problem says 'rotate about a given point.' Understanding the center is essential for describing symmetry — a regular hexagon, for example, has rotational symmetry about its center at angles of 60°, 120°, 180°, 240°, and 300°. In real-world applications, the concept appears in engineering (gears rotate about an axle), physics (torque acts about a pivot point), and computer graphics (sprites and objects rotate about specified anchor points).

Common Mistakes

Mistake: Applying the rotation formula without first translating to the center of rotation.

Correction: The standard rotation formulas x' = x cos θ − y sin θ and y' = x sin θ + y cos θ assume the center is at the origin. If the center is (h, k), you must first subtract (h, k), apply the rotation, and then add (h, k) back.

Mistake: Confusing the direction of rotation (clockwise vs. counterclockwise).

Correction: By convention, a positive angle θ means counterclockwise rotation. A clockwise rotation of θ is the same as a counterclockwise rotation of −θ. Mixing these up produces the wrong image coordinates.

Related Terms

Rotation — The transformation defined by a center and angle
Point — The center of rotation is a specific point
Plane — Rotation acts on points in the plane
Fixed — The center is the fixed (unmoved) point
Rotational Symmetry — A figure maps onto itself about its center
Transformation — Rotation is one of the four rigid transformations
Angle — The angle of rotation pairs with the center
Isometry — Rotation is a distance-preserving isometry