Regular Tessellation — Definition, Formula & Examples
A regular tessellation is a pattern that covers a flat surface entirely using copies of just one type of regular polygon, with no gaps or overlaps.
A regular tessellation is an edge-to-edge tiling of the Euclidean plane by congruent copies of a single regular polygon such that the same number of polygons meet at every vertex and the interior angles at each vertex sum to exactly .
Key Formula
Where:
- = Number of sides of the regular polygon
How It Works
For a regular polygon to tessellate the plane, its interior angle must divide evenly into . An equilateral triangle has interior angles of , and , so six triangles meet at each vertex. A square has angles, and , so four squares meet at each vertex. A regular hexagon has angles, and , so three hexagons meet at each vertex. No other regular polygon has an interior angle that divides evenly, which is why only these three regular tessellations exist.
Worked Example
Problem: Can a regular pentagon tessellate the plane?
Find the interior angle: A regular pentagon has 5 sides. Use the interior angle formula.
Check divisibility into 360°: Divide 360° by the interior angle to see if a whole number of pentagons can meet at a vertex.
Conclude: Since 3.33... is not a whole number, pentagons leave gaps at each vertex and cannot form a regular tessellation.
Answer: No. A regular pentagon cannot form a regular tessellation because its interior angle of does not divide evenly into .
Why It Matters
Regular tessellations appear in tile flooring, honeycomb structures, and graphic design. Understanding them strengthens your grasp of angle relationships in polygons, a topic tested in high school geometry and useful in architecture and computer graphics.
Common Mistakes
Mistake: Assuming any regular polygon can tessellate the plane if you try hard enough.
Correction: Only equilateral triangles, squares, and regular hexagons work. The interior angle must divide exactly into , and for polygons with 7 or more sides (and the pentagon), it never does.