Lesson 2
Regular Tessellations
Let’s make some regular tessellations.
2.1: Regular Tessellations
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For each shape (triangle, square, pentagon, hexagon, and octagon), decide if you can use that shape to make a regular tessellation of the plane. Explain your reasoning.
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For the polygons that do not work what goes wrong? Explain your reasoning.
2.2: Equilateral Triangle Tessellation
- What is the measure of each angle in an equilateral triangle? How do you know?
- How many triangles can you fit together at one vertex? Explain why there is no space between the triangles.
- Explain why you can continue the pattern of triangles to tessellate the plane.
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How can you use your triangular tessellation of the plane to show that regular hexagons can be used to give a regular tessellation of the plane?
2.3: Regular Tessellation for Other Polygons
![Five regular polygons. One 9-sided, one 10-sided, one 7-sided, one 8-sided, one 12-sided.](https://cms-im.s3.amazonaws.com/1ZPtNWvyPJiN5CfW1mAcMjie?response-content-disposition=inline%3B%20filename%3D%228-8.9.C1.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%278-8.9.C1.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T005754Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7ec389c9f427b7c6c0707fe793df048eaf4a6617b0aa2c7b4e57f5ba9262ac81)
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Can you make a regular tessellation of the plane using regular polygons with 7 sides? What about 9 sides? 10 sides? 11 sides? 12 sides? Explain.
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How does the measure of each angle in a square compare to the measure of each angle in an equilateral triangle? How does the measure of each angle in a regular 8-sided polygon compare to the measure of each angle in a regular 7-sided polygon?
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What happens to the angles in a regular polygon as you add more sides?
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Which polygons can be used to make regular tessellations of the plane?