Tessellating Shapes Worksheet | Fun and Engaging PDF Worksheets - Free Printable

This worksheet focuses on tessellating shapes, specifically triangles and quadrilaterals. Let's go through the tasks step-by-step, explain how to solve them, and provide guidance for completing each section.

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🔷 What is Tessellation?

Tessellation means covering a plane (like a flat surface) with shapes so that:
- There are no gaps or overlaps.
- The shapes fit together perfectly, edge-to-edge.
- The pattern can continue infinitely.

We'll explore whether all triangles and all quadrilaterals can tessellate.

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## ✔ Section A: Triangles

Task:

Show how each of the three given triangles tessellates by drawing at least six copies in the grid. Then investigate: *Can all triangles tessellate?*

#### Step-by-step guide:

1. Triangle 1 (Yellow Equilateral Triangle):
- This triangle has all sides equal and all angles = 60°.
- To tessellate: Place another triangle next to it so that edges match.
- You can rotate and reflect the triangle.
- Since 6 × 60° = 360°, six equilateral triangles meet at a point → perfect tessellation.
- Draw multiple copies around a central point or in a honeycomb-like pattern.

2. Triangle 2 (Pink Right-Angled Triangle):
- This is a right triangle (90° angle).
- Use the fact that two such triangles can form a rectangle.
- Then, you can tile the plane with rectangles.
- Or place them edge-to-edge, alternating orientation.
- Example: Place one triangle, then rotate it 180° and place it adjacent.
- You’ll find they fit perfectly without gaps.

3. Triangle 3 (Blue Scalene Triangle):
- All sides and angles are different.
- But any triangle can be used to make a parallelogram (by joining two identical triangles).
- Parallelograms tessellate easily.
- So even scalene triangles can tessellate!

#### Investigation: Can ALL triangles tessellate?

✔ Yes!
Every triangle can tessellate the plane.
Why? Because:
- Any triangle can be paired with itself to form a parallelogram.
- Parallelograms tessellate because their opposite sides are parallel and equal.
- By translating (sliding) the parallelogram repeatedly, you cover the plane.

> 🔎 Try this: Take any triangle, flip it over its side, and repeat. You’ll see the pattern continues seamlessly.

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📌 How to Draw on the Grid:

- Use the dot grid to align your shapes precisely.
- For each triangle:
- Copy the shape using the same size and orientation.
- Rotate or reflect as needed.
- Keep edges touching and no gaps.
- Draw at least six copies per triangle.

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## ✔ Section B: Quadrilaterals

Task:

Show how the three given quadrilaterals tessellate. Then test if all quadrilaterals can tessellate.

#### Step-by-step guide:

1. Green Trapezium (Quadrilateral 1):
- It has one pair of parallel sides.
- You can translate (slide) it horizontally and vertically.
- Place one trapezium, then slide it down and attach another.
- Rotate it 180° and place it beside to fill gaps.
- They will fit together like bricks in a wall.

2. Blue Kite-shaped Quadrilateral (Quadrilateral 2):
- This is a kite (two pairs of adjacent equal sides).
- Try rotating it around its center or vertices.
- Often, kites tessellate by forming star-like patterns or repeating in rows.
- You can use reflections or rotations to fill the space.

3. Red Irregular Quadrilateral (Quadrilateral 3):
- It looks like a "right-angled" irregular shape.
- Even if it’s not symmetric, try placing it and rotating it.
- Most irregular quadrilaterals can tessellate using rotation and translation.

#### Investigation: Can ALL quadrilaterals tessellate?

✔ Yes!
Any quadrilateral (convex or concave) can tessellate the plane.

Why?
- The sum of internal angles in any quadrilateral is 360°.
- When you place four copies of a quadrilateral around a point, their angles can add up to 360°.
- You can create a tiling by rotating the quadrilateral around its vertices or using translation.

> 🔎 Try this: Take any quadrilateral, copy it, rotate it 180° about the midpoint of a side, and repeat. This creates a tessellation.

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Final Task: Draw Your Own Quadrilateral and Tessellate It

- Choose any quadrilateral (e.g., a random four-sided shape).
- Draw it on the large grid below.
- Try to repeat it using translations, rotations, or reflections.
- If done correctly, it should fill the space without gaps or overlaps.

> 💡 Tip: Start by drawing the shape, then draw a rotated version next to it. Continue the pattern.

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## ✔ Summary of Answers:

| Shape Type | Can It Tessellate? | Why? |
|-----------|--------------------|------|
| All Triangles | ✔ Yes | Can be joined to form parallelograms; angles add up to 360° when repeated. |
| All Quadrilaterals | ✔ Yes | Sum of interior angles = 360°; can be arranged around a point or translated/rotated to fill plane. |

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## 🖼️ Visual Tips for Drawing:

- Use a ruler for straight lines.
- Match edges exactly.
- Use dots or grid lines to keep alignment.
- Label or color each shape to distinguish them.

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✔ Final Answer:

> Yes, all triangles and all quadrilaterals can tessellate.
> This is due to geometric properties:
> - Triangles: Two form a parallelogram, which tessellates.
> - Quadrilaterals: Their internal angles sum to 360°, allowing them to fit around a point.

Now go ahead and complete the grids by drawing at least six copies of each shape, showing how they fit together seamlessly! 🧩