Tiling — Definition, Formula & Examples
Tiling is a way of covering a flat surface completely using one or more shapes, with no gaps between them and no overlapping.
A tiling (or tessellation) of the plane is an arrangement of closed shapes that covers the entire plane without gaps or overlaps. The shapes used are called tiles, and each point on the surface belongs to at least one tile.
How It Works
To create a tiling, you choose one or more shapes and arrange copies of them so they fit together perfectly across the surface. Only three regular polygons can tile the plane by themselves: equilateral triangles, squares, and regular hexagons. Other shapes can also tile if their edges and angles combine to fill the space around every vertex. At each point where tiles meet, the angles must sum to exactly .
Worked Example
Problem: Can a regular pentagon (interior angle 108°) tile the plane by itself?
Step 1: Find the interior angle of a regular pentagon.
Step 2: Check whether copies of 108° can sum to exactly 360° at a vertex.
Step 3: Since 3.33... is not a whole number, you cannot fit a whole number of regular pentagons around a single point without leaving a gap or causing overlap.
Answer: No, a regular pentagon cannot tile the plane by itself because its interior angle does not divide evenly into 360°.
Why It Matters
Tiling shows up in architecture, art, and design — from bathroom floors to the geometric patterns in Islamic art. Understanding which shapes tessellate builds skills in angle reasoning and spatial thinking that carry into high school geometry and beyond.
Common Mistakes
Mistake: Assuming any regular polygon can tile the plane.
Correction: Only equilateral triangles, squares, and regular hexagons work among regular polygons. Always check that the interior angle divides evenly into 360°.