Rotation of Shapes

Rotation is a geometric transformation that turns every point of a figure about a fixed point (called the center of rotation) through a given angle and direction (clockwise or counterclockwise), without changing the shape's size or internal angles. In mathematical terms, rotation is an isometry: a transformation that preserves distances and shape properties.​

Mathematical Representation

1) Rotation about Origin

P'(x', y') = (x cosθ - y sinθ, x sinθ + y cosθ)

2) Quick Rules (Standard Angles)

Types of Rotation

1. Based on Direction

• Counterclockwise (Positive angle θ): Standard mathematical convention

• Clockwise (Negative angle {-θ}): Opposite to standard direction

2. Based on Center

• Origin Rotation: Center at (0,0)

• Arbitrary Point Rotation: Center at C(cx,cy)

3. Based on Dimension

• 2D Rotation: Plane rotation (x,y coordinates)

• 3D Rotation: Space rotation (x,y,z coordinates)

Clockwise & Counterclockwise Rotation

Counterclockwise Rotation :
Rotation in the positive angle direction (standard mathematical convention) where points move leftward from the positive x-axis around the center. Every point traces a counterclockwise circular arc, preserving distance from rotation center.

Clockwise Rotation :
Rotation in the negative angle direction (opposite to standard convention) where points move rightward from the positive x-axis around the center. Equivalent to counterclockwise rotation by -θ.

Origin Rotation & Arbitrary Point Rotation

Origin Rotation : Rotation about the fixed point (0,0) using simple coordinate transformation rules. Every point rotates around the origin maintaining constant distance from (0,0).

Arbitrary Point Rotation : Rotation about any point C(cx,cy) (not origin) requires 3-step process:

(1) Translate so C becomes origin

(2) Rotate about new origin

(3) Translate back to original position.

All points maintain same distance from C after rotation.

Step 1: Translate: P(x,y) → P₁ (x-cx, y-cy)

Step 2: Rotate: P₁ → P₂ (x₁cosθ - y₁sinθ, x₁sinθ + y₁cosθ)

Step 3: Translate back: P₂ → P' (x₂ + cx, y₂ + cy)

Example: Rectangle Rotation 90° Counterclockwise about C(2, 2)

Original Rectangle ABCD: A(1, 1), B(4, 1), C(2, 2), D(2, 4) rotated 90° Counterclockwise about center C(2, 2)

Solution :-

Step 1: Translate C(2,2) → Origin

A(1,1) → A₁(1-2,1-2) = (-1,-1)

B(4,1) → B₁(4-2,1-2) = (2,-1)

D(2,4) → D₁(2-2,4-2) = (0,2)

Step 2: 90° CCW Rotation [(-y,x) Rule]

A₁(-1,-1) → A₂(1,-1)

B₁(2,-1) → B₂(1,2)

D₁(0,2) → D₂(-2,0)

Step 3: Translate Back C(2,2)

A₂(1,-1) → A'(1+2,-1+2) = (3,1)

B₂(1,2) → B'(1+2,2+2) = (3,4)

D₂(-2,0) → D'(-2+2,0+2) = (0,2)

Final Answer: A'(3,1), B'(3,4), D'(0,2)

2D & 3D Rotation

2D Rotation : It transforms points in a plane (x,y coordinates) around a fixed center by angle θ. Points move along circular arcs maintaining constant distance from rotation center.

2D Rotation Matrix about Origin:

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

3D Rotation : It transforms points in space (x,y,z coordinates) around a fixed axis (X, Y, or Z) by angle θ.

Homogeneous Coordinates (3×3 for composition):

\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}

Rotation of Shape - Solved Question and Answers

Question 1 : Point P(3,−2) is rotated 90° counterclockwise about the origin. Find the coordinates of P′ .

Solution :

For 90° CounterClockWise about origin, use rule: (x , y) → (−y , x)

P(3 , −2) → P′ (−(−2) , 3) = (2,3)

Question 2: Point A(5,7) is rotated 180° about the origin. Find the coordinates of A′ .

Solution :

For 180° (same for Clockwise/Counterclockwise): (x,y) → (−x,−y).​

A(5 , 7) → A′(−5 , −7)

Answer:  A′(−5 , −7)

Question 3: The given figure is rotated 90° counterclockwise about the black dot (which acts as the center of rotation). Which of the following options correctly represents the rotated figure?

Solution:

• During a 90° counterclockwise rotation, the black dot remains fixed as the center of rotation.
• All parts of the figure move along circular paths, maintaining their distances and orientation.
• Only Option (C) shows the correct position and orientation after this rotation.

Realated Articles

• Assembling
• Grouping
• Paper Folding
• Cutting

Practice Problems on Rotation of Shapes

Question 1 : A triangle ABC with vertices A(1, 0), B(2, 0), and C(1, 1) is rotated 180° counterclockwise about the origin. What are the coordinates of the rotated triangle A'B'C'?

Question 2 : Point P(6,3) is rotated 90° counterclockwise about point C(2,2) . Find P′.

Question 3 : Point P(2,−3) is rotated:

• 90° counterclockwise about origin
• 270° counterclockwise about origin . Find the coordinates after each rotation.

Question 4 : Triangle T1 (1,1),(4,1),(2,3) becomes T2 (-1,-1),(-4,-1),(-2,-3). Is T2 a rotation of T1 ? If yes, what angle and center?