Rotational Symmetry All 2 dimensional shapes have some rotational ... - Free Printable

Let's solve the problem of finding the order of rotational symmetry for each shape.

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🔷 What is Rotational Symmetry?

A shape has rotational symmetry if it looks the same after being rotated by a certain angle around its center. The order of rotational symmetry is the number of times the shape looks exactly the same during a full 360° rotation.

For example:
- A square looks the same 4 times in a full rotation → order = 4.
- A circle looks the same at every angle → infinite order, but we usually say "infinite" or just note that it’s not applicable here.

We’ll go through each shape one by one.

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✔ Solutions:

#### 1. Star (Star of David / Hexagram)
- This is a six-pointed star made of two overlapping equilateral triangles.
- It matches itself every 60° (360° ÷ 6).
- So, it has 6 positions where it looks the same.
- Order: 6

#### 2. Quartered Circle (Half black, half white, quartered)
- Divided into four quadrants: alternating black and white.
- After rotating 90°, it looks the same.
- It matches every 90° → 360° ÷ 90° = 4 rotations.
- Order: 4

#### 3. Triangular Design (Concentric Triangles)
- Three nested equilateral triangles pointing the same way.
- Rotating by 120° brings it back to original appearance.
- 360° ÷ 120° = 3 → matches 3 times.
- Order: 3

#### 4. Hourglass-like Shape with Dots
- Looks like a bowtie with a central dot and symmetrical bars.
- Has two-fold symmetry: 180° rotation maps it onto itself.
- After 180°, it looks the same; at 360° again.
- No match at 90° or 270°.
- Order: 2

#### 5. Circle with Diagonal Arrow
- The arrow points in a specific direction.
- Only when rotated 360° does it look the same.
- At any smaller rotation, the arrow changes position.
- No rotational symmetry except full turn.
- Order: 1 (only identity)

#### 6. Three-armed Symbol (Like a Triforce or Propeller)
- Three identical arms arranged symmetrically.
- Matches every 120° → 360° ÷ 120° = 3
- Order: 3

#### 7. Square
- Standard square.
- Matches at 90°, 180°, 270°, 360° → 4 positions.
- Order: 4

#### 8. Five Squares Around a Circle
- Five squares equally spaced around a circle, each with a dot.
- Each rotation of 72° (360° ÷ 5) gives same appearance.
- Order: 5

#### 9. Square with Inner Cross and Circles
- Outer square, inner cross, and circles at intersections.
- Looks the same at 90°, 180°, 270°, 360°.
- Symmetric under 4 rotations.
- Order: 4

#### 10. Spiral Pattern
- A spiral with concentric curves.
- No matter how you rotate it, the spiral appears different unless you do 360°.
- Only matches at full rotation.
- Order: 1

#### 11. Snowflake (Six-Pointed)
- Classic snowflake design with 6-fold symmetry.
- Matches every 60° → 360° ÷ 60° = 6
- Order: 6

#### 12. Octagon (Regular Octagon)
- Regular 8-sided polygon.
- Matches every 45° → 360° ÷ 45° = 8
- Order: 8

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✔ Final Answers:

| Number | Shape Description | Order of Rotational Symmetry |
|--------|---------------------------------------|-------------------------------|
| 1 | Six-pointed star | 6 |
| 2 | Quartered circle (alternating) | 4 |
| 3 | Concentric triangles | 3 |
| 4 | Bowtie with dots | 2 |
| 5 | Circle with arrow | 1 |
| 6 | Three-armed symbol | 3 |
| 7 | Square | 4 |
| 8 | Five squares around a circle | 5 |
| 9 | Square with inner cross | 4 |
| 10 | Spiral pattern | 1 |
| 11 | Snowflake | 6 |
| 12 | Regular octagon | 8 |

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📌 Summary:

Rotational symmetry depends on how many times a shape maps onto itself during a full rotation. Shapes with uniform repetition (like regular polygons or stars) have higher orders, while asymmetric or directional elements (like arrows or spirals) often have order 1.

Let me know if you'd like visual explanations or diagrams!