Tiling and Tessellation
Look at the floor of your classroom or bathroom. The tiles fit together perfectly with no gaps and no overlaps. This arrangement is called a tessellation (or tiling).
In Class 4, you will learn which shapes can tessellate, why some shapes leave gaps, and how to create your own tiling patterns. This topic connects geometry with art and design.
What is Tiling and Tessellation - Class 4 Maths (Patterns)?
A tessellation (or tiling) is a pattern of shapes that:
- Covers a flat surface completely
- Has no gaps between shapes
- Has no overlaps of shapes
The shapes can be the same (regular tessellation) or a mix of different shapes (semi-regular tessellation).
Tessellation = No gaps + No overlaps + Covers the entire surface
Types and Properties
Shapes that tessellate:
| Shape | Tessellates? | Reason |
|---|---|---|
| Square | Yes | 4 squares meet at each point (4 × 90° = 360°) |
| Equilateral Triangle | Yes | 6 triangles meet at each point (6 × 60° = 360°) |
| Regular Hexagon | Yes | 3 hexagons meet at each point (3 × 120° = 360°) |
| Rectangle | Yes | Works like squares with adjusted dimensions |
| Regular Pentagon | No | 3 × 108° = 324° (not 360°, leaves gaps) |
| Circle | No | Curved edges always leave gaps |
Rule: At each meeting point, the angles of the shapes must add up to exactly 360° for a perfect tessellation.
Solved Examples
Example 1: Example 1: Square tiles
Problem: Can square tiles cover a floor with no gaps or overlaps?
Solution:
Step 1: Each angle of a square = 90°.
Step 2: At each meeting point: 4 squares meet → 4 × 90° = 360°.
Step 3: 360° means a complete turn — no gap.
Answer: Yes, squares tessellate perfectly.
Example 2: Example 2: Why circles don't tessellate
Problem: Meera tries to tile a wall with circular shapes. Will it work?
Solution:
Step 1: Circles have curved edges.
Step 2: No matter how you arrange circles, there are gaps between them.
Answer: No, circles cannot tessellate because curved edges always leave gaps.
Example 3: Example 3: Equilateral triangles
Problem: Do equilateral triangles tessellate?
Solution:
Step 1: Each angle = 60°.
Step 2: At each point: 6 triangles meet → 6 × 60° = 360° ✓
Answer: Yes, equilateral triangles tessellate. Six triangles fit around each point.
Example 4: Example 4: Regular hexagons
Problem: A honeycomb is made of regular hexagons. Why does this shape tessellate?
Solution:
Step 1: Each interior angle of a regular hexagon = 120°.
Step 2: At each vertex: 3 hexagons meet → 3 × 120° = 360° ✓
Answer: Regular hexagons tessellate because 3 of them meet at each point to give exactly 360°.
Example 5: Example 5: Why regular pentagons don't tessellate
Problem: Can regular pentagons tile a floor?
Solution:
Step 1: Each interior angle = 108°.
Step 2: 3 pentagons: 3 × 108° = 324° (36° short of 360° — leaves a gap).
Step 3: 4 pentagons: 4 × 108° = 432° (72° too much — would overlap).
Answer: No, regular pentagons cannot tessellate because their angles cannot add to exactly 360° at any meeting point.
Example 6: Example 6: Rectangles
Problem: Aman uses rectangular tiles (each 20 cm × 10 cm) for his bathroom floor. Will they tessellate?
Solution:
Step 1: A rectangle has 4 right angles (90° each).
Step 2: 4 rectangles meet at each corner: 4 × 90° = 360° ✓
Answer: Yes, rectangles tessellate perfectly, just like squares.
Example 7: Example 7: Combining two shapes
Problem: Can squares and equilateral triangles be combined to make a tessellation?
Solution:
Step 1: Square angle = 90°. Triangle angle = 60°.
Step 2: Try: 90 + 90 + 60 + 60 + 60 = 360° ✓ (2 squares + 3 triangles at a point).
Answer: Yes, this combination can tessellate. This is a semi-regular tessellation.
Example 8: Example 8: Tiling a rectangular floor
Problem: A floor is 200 cm × 150 cm. How many 50 cm × 50 cm square tiles are needed?
Solution:
Step 1: Tiles along length: 200 ÷ 50 = 4
Step 2: Tiles along width: 150 ÷ 50 = 3
Step 3: Total tiles: 4 × 3 = 12
Answer: 12 tiles are needed.
Example 9: Example 9: Irregular shape that tessellates
Problem: Can any quadrilateral (4-sided shape) tessellate, even if it is not regular?
Solution:
Step 1: The angles of any quadrilateral add up to 360°.
Step 2: If we place copies of the quadrilateral so that all four different angles meet at one point, they add to 360°.
Answer: Yes, any quadrilateral (regular or irregular) can tessellate.
Example 10: Example 10: Real-life tessellation
Problem: Aditi sees a brick wall. The bricks are rectangles arranged in a staggered pattern. Is this a tessellation?
Solution:
Step 1: The bricks cover the wall completely.
Step 2: No gaps between bricks. No overlaps.
Answer: Yes, a brick wall is a tessellation. The staggered arrangement (offset rows) is a common tiling pattern.
Key Points to Remember
- A tessellation covers a surface with no gaps and no overlaps.
- Only three regular polygons tessellate on their own: equilateral triangles, squares, and regular hexagons.
- The angles at each meeting point must add up to exactly 360°.
- Any triangle and any quadrilateral can tessellate (regular or irregular).
- Circles and regular pentagons cannot tessellate.
- Different shapes can be combined to create semi-regular tessellations.
- Real-life examples: floor tiles, brick walls, honeycombs, jigsaw puzzles, rangoli patterns.
Practice Problems
- Name the three regular polygons that can tessellate by themselves.
- Can regular octagons (8 sides) tessellate alone? Why or why not? (Interior angle = 135°)
- A floor is 300 cm × 200 cm. How many 25 cm × 25 cm tiles are needed?
- Why can circles not tessellate?
- Draw a tessellation using only equilateral triangles on a grid.
- Can a combination of regular hexagons and equilateral triangles tessellate? Check the angles.
- Identify two tessellation patterns you can see in your school.
Frequently Asked Questions
Q1. What is a tessellation?
A tessellation is a pattern of shapes that covers a flat surface completely with no gaps and no overlaps. Floor tiling is the most common real-life example.
Q2. Which regular shapes can tessellate?
Only three regular polygons can tessellate alone: equilateral triangles (6 fit at a point), squares (4 fit at a point), and regular hexagons (3 fit at a point).
Q3. Why must the angles add up to 360°?
A full turn around any point is 360°. If the shapes around a point add up to less than 360°, there is a gap. If they add up to more, the shapes overlap. Exactly 360° means a perfect fit.
Q4. Can irregular shapes tessellate?
Yes. Any triangle and any quadrilateral can tessellate, even if their sides and angles are not equal. The key is that their angle sums allow them to fit together at 360°.
Q5. Why can't a regular pentagon tessellate?
A regular pentagon has interior angles of 108°. Three pentagons give 324° (gap of 36°), and four give 432° (overlap). Since no combination of 108° angles makes exactly 360°, pentagons cannot tessellate.
Q6. What is a semi-regular tessellation?
A semi-regular tessellation uses two or more different regular polygons. At each vertex, the same combination of shapes meets. For example, squares and equilateral triangles can form a semi-regular tessellation.
Q7. Where do we see tessellations in real life?
Floor tiles, brick walls, bathroom mosaics, honeycomb, pineapple skin, rangoli designs, fabric patterns, and jigsaw puzzles are all examples of tessellations.
Q8. How do you calculate the number of tiles needed for a floor?
Divide the floor length by the tile length to get tiles per row. Divide the floor width by the tile width to get the number of rows. Multiply the two numbers for the total tiles needed.
