Worksheet for identifying tessellating 2D shapes.

The image you've uploaded is a "Which Shapes Tessellate?" worksheet (Sheet 1) from Math-Worksheets4Kids.com, designed to help students understand which shapes can tessellate — meaning they can cover a plane with no gaps or overlaps using only translations, rotations, or reflections of the same shape.

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🔍 Task:

Identify which of the given shapes can tessellate (tile the plane without gaps or overlaps).
You are instructed: *"Remember: you can only use the shape on its own to cover each pattern."*

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✔ Step-by-step Explanation:

Let’s go through each shape one by one and determine if it tessellates.

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#### 1. Right Triangle (Top Left)
- ✔ Yes, it tessellates.
- Any triangle can tessellate because triangles can be arranged in a repeating pattern to fill a plane.
- For example, two right triangles can form a rectangle, and rectangles tessellate.
- So, even alone, right triangles can be rotated and reflected to tile a surface.

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#### 2. Pentagon (Top Middle)
- ✘ No, it does not tessellate.
- A regular pentagon cannot tessellate the plane because its internal angles (108°) do not divide evenly into 360°.
- When you try to place them around a point, there will be gaps or overlaps.
- Some irregular pentagons *can* tessellate, but this appears to be a regular pentagon, so it does not tessellate.

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#### 3. Circle (Top Right)
- ✘ No, it does not tessellate.
- Circles cannot tessellate because they leave gaps between them when placed next to each other.
- There is no way to arrange circles to cover a plane without gaps — unless you allow overlapping, which isn’t allowed in tessellation.

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#### 4. Oval (Middle Left)
- ✘ No, it does not tessellate.
- Like circles, ovals have curved edges that prevent them from fitting together without gaps.
- No matter how you rotate or reflect an oval, there will always be empty spaces.

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#### 5. Arrow-like Shape (Middle Center)
- ✔ Yes, it tessellates.
- This is a polygon with straight sides and sharp angles.
- Many irregular polygons can tessellate if their angles add up appropriately.
- This shape resembles a "chevron" or "arrowhead," and such shapes often tessellate due to symmetry.
- You can imagine rotating and translating this shape to fill space.

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#### 6. Rectangle (Middle Right)
- ✔ Yes, it tessellates.
- Rectangles are classic examples of tessellating shapes.
- They can be repeated horizontally and vertically to cover a plane perfectly.
- Even non-square rectangles tessellate.

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#### 7. House-shaped Pentagon (Bottom Left)
- ✔ Yes, it tessellates.
- This is an irregular pentagon shaped like a house (square with a triangle on top).
- While regular pentagons don't tessellate, this specific shape can be repeated and rotated to form a tiling pattern.
- Think of brick patterns or roof tiles — similar shapes can fit together.
- So yes, this shape can tessellate.

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#### 8. Wavy Flag-like Shape (Bottom Middle)
- ✘ No, it does not tessellate.
- This shape has curved edges and an irregular wavy side.
- The curves mean that when you try to place adjacent copies, there will be gaps or overlaps.
- Also, the asymmetry makes it hard to repeat seamlessly.
- So, this cannot tessellate.

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#### 9. Star (Bottom Right)
- ✘ No, it does not tessellate.
- A regular star (like a 5-pointed star) cannot tessellate the plane on its own.
- Its points and angles create gaps when placed together.
- While stars can be part of larger tiling designs, a single star shape cannot tile the plane without gaps or overlaps.
- So, no, it does not tessellate.

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✔ Final Answer:

Shapes that tessellate:
1. Right Triangle ✔
2. Arrow-like Shape ✔
3. Rectangle ✔
4. House-shaped Pentagon ✔

Shapes that do NOT tessellate:
1. Regular Pentagon ✘
2. Circle ✘
3. Oval ✘
4. Wavy Flag ✘
5. Star ✘

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📝 Summary:

Only four of the nine shapes can tessellate:
- Right Triangle
- Arrow-like Shape
- Rectangle
- House-shaped Pentagon

These are all polygons with straight sides and angles that allow them to fit together perfectly.

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✔ Answer Key (Tessellating Shapes):
- ✔ Right Triangle
- ✔ Arrow-like Shape
- ✔ Rectangle
- ✔ House-shaped Pentagon

All others: ✘

Let me know if you'd like a visual explanation or printable answer key!