Rotation of Shapes (original) (raw)

Last Updated : 24 Dec, 2025

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?

Question 3

**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?