Lesson 6.4: Tessellations & Patterns

Learning Objectives

• Define tessellations and identify tessellating shapes.
• Understand how interior and exterior angles influence tiling.
• Create and analyze repeating geometric patterns.

Topics Covered

• Definition of tessellation
• Regular polygons and tiling rules
• Interior and exterior angles in tessellations
• Creating repeating patterns
• Applications in art and architecture

Key Terms

• Tessellation: A pattern of shapes that completely covers a plane without gaps or overlaps.
• Regular Polygon: A polygon with all sides and angles equal.
• Interior Angle: The angle formed inside a polygon.
• Exterior Angle: The angle formed outside a polygon, supplementary to the interior angle.
• Edge-to-Edge Tessellation: Tessellation where sides of polygons coincide exactly.

Lesson Examples

Example 1: Tessellation with Squares

1. Each interior angle of a square is 90°.
2. Four squares meet at a point: 4 × 90° = 360° → fits perfectly → tessellation achieved.

Example 2: Tessellation with Equilateral Triangles

1. Each interior angle of an equilateral triangle is 60°.
2. Six triangles meet at a point: 6 × 60° = 360° → plane covered → tessellation achieved.

Example 3: Hexagon Tessellation

1. Each interior angle of a regular hexagon is 120°.
2. Three hexagons meet at a point: 3 × 120° = 360° → tessellation works.

Practice Problems

1. Determine if regular pentagons can tessellate a plane.
2. Create a tessellation pattern using equilateral triangles and squares.
3. Calculate the number of regular octagons meeting at a point for a tessellation.

Tips & Common Mistakes

• Check that the sum of angles around a point equals 360° for tessellation.
• Not all regular polygons tessellate the plane (e.g., pentagons cannot alone tessellate).
• Use edge-to-edge alignment for accurate patterns.

Summary

Tessellations cover planes using repeated shapes without gaps. Only triangles, squares, and hexagons among regular polygons can tessellate alone. Understanding interior and exterior angles is key to analyzing patterns.

Challenge Problems

1. Design a tessellation combining hexagons and triangles. Explain the angles meeting at each vertex.
2. Explain why a regular pentagon cannot tessellate the plane by itself.
3. Create a tessellation pattern using squares and equilateral triangles; calculate angles at shared vertices.