Tessellations in Geometry 1 - Free Printable
Educational worksheet: Tessellations in Geometry 1. Download and print for classroom or home learning activities.
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Step-by-step solution for: Tessellations in Geometry 1
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Show Answer Key & Explanations
Step-by-step solution for: Tessellations in Geometry 1
It looks like you've uploaded a worksheet titled "Which Shapes Tessellate? Sheet 1" from Math-Salamanders.com, which asks students to determine which of the given shapes can tessellate (i.e., cover a plane without gaps or overlaps using only translations, rotations, or reflections).
Let’s go through each shape one by one and analyze whether it can tessellate.
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Tessellation means that a shape can be repeated in a pattern to cover a flat surface completely with no gaps or overlaps. The key idea is that the shapes must fit together perfectly around a point, typically requiring that the angles around a vertex sum to 360°.
We can use regular polygons as a guide:
- Regular triangles (equilateral): Yes – 60° angles → 6 × 60° = 360°
- Squares: Yes – 90° angles → 4 × 90° = 360°
- Regular hexagons: Yes – 120° angles → 3 × 120° = 360°
- Regular pentagons: No – 108° → doesn't divide evenly into 360°
But not all shapes need to be regular — irregular shapes can also tessellate if they fit together properly.
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Now let's evaluate each shape:
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- This is a right triangle.
- Any triangle can tessellate.
- Why? You can place two identical triangles together to form a parallelogram, and parallelograms tessellate.
- So, YES, this triangle can tessellate.
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- A trapezoid has one pair of parallel sides.
- Many trapezoids can tessellate, especially isosceles trapezoids.
- By rotating and translating, you can create a repeating pattern.
- YES, this shape can tessellate.
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- Circles cannot tessellate.
- There will always be gaps between circles when placed next to each other.
- Even though they can pack tightly (like in a honeycomb), they don’t tile the plane without gaps.
- NO, circles do not tessellate.
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- Similar to a circle, an oval cannot tessellate.
- It has curved edges, so there will always be gaps between them.
- NO, ovals do not tessellate.
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- Rectangles are a type of quadrilateral.
- They can tessellate easily — just repeat them side-by-side.
- Think of floor tiles or bricks.
- YES, this rectangle can tessellate.
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- This is an irregular pentagon.
- But notice: it looks like a "house" shape or a chevron.
- This kind of shape can tessellate — many irregular polygons can.
- For example, pentominoes (shapes made of 5 squares) often tessellate.
- This particular shape may be designed to fit together with rotations and translations.
- YES, this shape can tessellate.
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- This is a composite shape: a rectangle with a triangle on top.
- This is a common tessellating shape used in tiling patterns.
- You can see this in brickwork or tile designs.
- YES, this shape can tessellate.
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- This shape has curved edges.
- However, if it's a symmetrical wave (like a sine wave), it might still tessellate.
- But in general, curved shapes do not tessellate unless specifically designed.
- However, this shape appears to be a rectangle with wavy sides, and if it's a "bump" and "dip" design, it might fit with another copy.
- In fact, such "wave" shapes can tessellate if the waves are complementary.
- Since this is likely a design intended for tessellation, and it matches a common pattern, YES, it can tessellate.
> ⚠️ Note: This depends on symmetry. If the left side is a bump and the right is a dip, then flipping and placing next to each other allows tessellation.
So, assuming it's symmetric and designed to interlock, YES, it can tessellate.
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- A regular five-pointed star (like the one on the American flag) does not tessellate on its own.
- The angles at the points and indentations don't allow perfect fitting without gaps.
- While stars can be part of a larger tessellation pattern (e.g., with other shapes), a single star shape cannot fill a plane without gaps.
- NO, this star does not tessellate.
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| Shape | Can it Tessellate? | Reason |
|------|--------------------|--------|
| 1. Right Triangle | ✔ Yes | All triangles tessellate |
| 2. Trapezoid | ✔ Yes | Parallelograms and trapezoids tessellate |
| 3. Circle | ✘ No | Curved edges leave gaps |
| 4. Oval | ✘ No | Same reason as circle |
| 5. Rectangle | ✔ Yes | Standard tiling shape |
| 6. Arrow-shaped Pentagon | ✔ Yes | Irregular but fits well |
| 7. House Shape | ✔ Yes | Common tessellating composite shape |
| 8. Wavy Flag | ✔ Yes | Designed to interlock |
| 9. Star | ✘ No | Angles don’t add up to 360° |
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The shapes that can tessellate are:
- 1 (Triangle)
- 2 (Trapezoid)
- 5 (Rectangle)
- 6 (Arrow)
- 7 (House)
- 8 (Wavy Flag)
The shapes that cannot tessellate are:
- 3 (Circle)
- 4 (Oval)
- 9 (Star)
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✔ Answer:
Shapes that tessellate: 1, 2, 5, 6, 7, 8
Shapes that do NOT tessellate: 3, 4, 9
Let me know if you'd like a visual explanation or how to draw the tessellations!
Let’s go through each shape one by one and analyze whether it can tessellate.
---
🔍 What is Tessellation?
Tessellation means that a shape can be repeated in a pattern to cover a flat surface completely with no gaps or overlaps. The key idea is that the shapes must fit together perfectly around a point, typically requiring that the angles around a vertex sum to 360°.
We can use regular polygons as a guide:
- Regular triangles (equilateral): Yes – 60° angles → 6 × 60° = 360°
- Squares: Yes – 90° angles → 4 × 90° = 360°
- Regular hexagons: Yes – 120° angles → 3 × 120° = 360°
- Regular pentagons: No – 108° → doesn't divide evenly into 360°
But not all shapes need to be regular — irregular shapes can also tessellate if they fit together properly.
---
Now let's evaluate each shape:
---
✔ Shape 1: Right Triangle
- This is a right triangle.
- Any triangle can tessellate.
- Why? You can place two identical triangles together to form a parallelogram, and parallelograms tessellate.
- So, YES, this triangle can tessellate.
---
✔ Shape 2: Trapezoid
- A trapezoid has one pair of parallel sides.
- Many trapezoids can tessellate, especially isosceles trapezoids.
- By rotating and translating, you can create a repeating pattern.
- YES, this shape can tessellate.
---
✘ Shape 3: Circle
- Circles cannot tessellate.
- There will always be gaps between circles when placed next to each other.
- Even though they can pack tightly (like in a honeycomb), they don’t tile the plane without gaps.
- NO, circles do not tessellate.
---
✔ Shape 4: Oval (Ellipse)
- Similar to a circle, an oval cannot tessellate.
- It has curved edges, so there will always be gaps between them.
- NO, ovals do not tessellate.
---
✔ Shape 5: Rectangle
- Rectangles are a type of quadrilateral.
- They can tessellate easily — just repeat them side-by-side.
- Think of floor tiles or bricks.
- YES, this rectangle can tessellate.
---
✔ Shape 6: Arrow-like Shape (Pentagon with a "tail")
- This is an irregular pentagon.
- But notice: it looks like a "house" shape or a chevron.
- This kind of shape can tessellate — many irregular polygons can.
- For example, pentominoes (shapes made of 5 squares) often tessellate.
- This particular shape may be designed to fit together with rotations and translations.
- YES, this shape can tessellate.
---
✔ Shape 7: House-like Shape (Triangle on top of a rectangle)
- This is a composite shape: a rectangle with a triangle on top.
- This is a common tessellating shape used in tiling patterns.
- You can see this in brickwork or tile designs.
- YES, this shape can tessellate.
---
✔ Shape 8: Wavy Flag Shape
- This shape has curved edges.
- However, if it's a symmetrical wave (like a sine wave), it might still tessellate.
- But in general, curved shapes do not tessellate unless specifically designed.
- However, this shape appears to be a rectangle with wavy sides, and if it's a "bump" and "dip" design, it might fit with another copy.
- In fact, such "wave" shapes can tessellate if the waves are complementary.
- Since this is likely a design intended for tessellation, and it matches a common pattern, YES, it can tessellate.
> ⚠️ Note: This depends on symmetry. If the left side is a bump and the right is a dip, then flipping and placing next to each other allows tessellation.
So, assuming it's symmetric and designed to interlock, YES, it can tessellate.
---
✘ Shape 9: Star (Five-pointed star)
- A regular five-pointed star (like the one on the American flag) does not tessellate on its own.
- The angles at the points and indentations don't allow perfect fitting without gaps.
- While stars can be part of a larger tessellation pattern (e.g., with other shapes), a single star shape cannot fill a plane without gaps.
- NO, this star does not tessellate.
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✔ Summary Table:
| Shape | Can it Tessellate? | Reason |
|------|--------------------|--------|
| 1. Right Triangle | ✔ Yes | All triangles tessellate |
| 2. Trapezoid | ✔ Yes | Parallelograms and trapezoids tessellate |
| 3. Circle | ✘ No | Curved edges leave gaps |
| 4. Oval | ✘ No | Same reason as circle |
| 5. Rectangle | ✔ Yes | Standard tiling shape |
| 6. Arrow-shaped Pentagon | ✔ Yes | Irregular but fits well |
| 7. House Shape | ✔ Yes | Common tessellating composite shape |
| 8. Wavy Flag | ✔ Yes | Designed to interlock |
| 9. Star | ✘ No | Angles don’t add up to 360° |
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✔ Final Answer:
The shapes that can tessellate are:
- 1 (Triangle)
- 2 (Trapezoid)
- 5 (Rectangle)
- 6 (Arrow)
- 7 (House)
- 8 (Wavy Flag)
The shapes that cannot tessellate are:
- 3 (Circle)
- 4 (Oval)
- 9 (Star)
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✔ Answer:
Shapes that tessellate: 1, 2, 5, 6, 7, 8
Shapes that do NOT tessellate: 3, 4, 9
Let me know if you'd like a visual explanation or how to draw the tessellations!
Parent Tip: Review the logic above to help your child master the concept of tessellations worksheet.
