Rotation Formula

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

The Formula

90∘,  180∘,  270∘ turns90^\circ,\;180^\circ,\;270^\circ\text{ turns}

When to use: Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Quick Example

Rotate 90°90° counterclockwise about origin: (1,0)→(0,1)(1, 0) \to (0, 1)

Notation

A rotation needs a center, angle, and direction.

What This Formula Means

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Formal View

Rθ:R2→R2R_\theta: \mathbb{R}^2 \to \mathbb{R}^2, Rθ(x,y)=(xcos⁡θ−ysin⁡θ,  xsin⁡θ+ycos⁡θ)R_\theta(x, y) = (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta); matrix form: Rθ=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, det⁡(Rθ)=1\det(R_\theta) = 1

Worked Examples

Example 1

easy
Rotate the point A(3,2)A(3, 2) by 90°90° counterclockwise about the origin. Where does A end up?

Answer

A′=(−2,3)A' = (-2, 3)

First step

1
Step 1: The rule for 90° CCW rotation is (x,y)→(−y,x)(x, y) \to (-y, x).

Full solution

  1. 2
    Step 2: Apply to A(3,2)A(3, 2): A′=(−2,3)A' = (-2, 3).
  2. 3
    Step 3: Verify distance from origin is unchanged: 32+22=13\sqrt{3^2+2^2} = \sqrt{13} and (−2)2+32=13\sqrt{(-2)^2+3^2} = \sqrt{13}. ✓
The 90° CCW rotation rule (x,y)→(−y,x)(x,y)\to(-y,x) is derived from the rotation matrix with θ=90°\theta=90°: cos⁡90°=0\cos90°=0, sin⁡90°=1\sin90°=1, giving (−y,x)(-y, x). Rotation preserves distance from the center of rotation.

Example 2

medium
Rotate the point B(4,0)B(4, 0) by 60°60° counterclockwise about the origin. Give exact coordinates.

Example 3

easy
Rotate the segment from A(1,2)A(1, 2) to B(4,2)B(4, 2) by 180°180° about the origin. Find A′B′A'B'.

Common Mistakes

  • Rotating around the wrong center — the same angle around a different center gives a different image.
  • Mixing clockwise and counterclockwise — direction matters except for 180 degrees.
  • Changing size — rotations preserve lengths and angle measures.

Why This Formula Matters

Rotations develop angle, coordinate, and congruence reasoning. They also force students to track center and direction, not just final appearance. Recognizing it by "What is the center, angle, and direction of the turn?" — rather than by familiar numbers — is what lets a student tell it apart from translation and reflection in a mixed problem set.

Frequently Asked Questions

What is the Rotation formula?

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

How do you use the Rotation formula?

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

What do the symbols mean in the Rotation formula?

A rotation needs a center, angle, and direction.

Why is the Rotation formula important in Math?

Rotations develop angle, coordinate, and congruence reasoning. They also force students to track center and direction, not just final appearance. Recognizing it by "What is the center, angle, and direction of the turn?" — rather than by familiar numbers — is what lets a student tell it apart from translation and reflection in a mixed problem set.

What do students get wrong about Rotation?

The procedure for rotation is the easy part; the trap is rotating around the wrong center. Asking "What is the center, angle, and direction of the turn?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Rotation formula?

Before studying the Rotation formula, you should understand: transformation geo, angles.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Geometry Transformations and Cross-Sections Guide →
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