Tiling Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tiling Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Covering an entire surface with copies of one or more shapes that fit together perfectly with no gaps and no overlaps.

Bathroom tiles cover the floor perfectly—no gaps between them.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: For a regular polygon to tile the plane alone, the interior angles at each meeting vertex must sum to exactly 360°.

Common stuck point: Only certain regular polygons can tile alone (3, 4, or 6 sides).

Sense of Study hint: Try placing copies of the shape around a single point. If the angles at that vertex add to exactly 360 degrees, the shape can tile.

Worked Examples

Example 1

easy
Can regular pentagons tile the plane by themselves (no gaps or overlaps)? Explain using interior angles.

Solution

  1. 1
    Step 1: Interior angle of a regular pentagon: \dfrac{(5-2) \times 180°}{5} = \dfrac{540°}{5} = 108°.
  2. 2
    Step 2: For a tiling to work without gaps, the angles meeting at each vertex must sum to exactly 360°.
  3. 3
    Step 3: 360° \div 108° = 3.33..., which is not a whole number. So pentagons cannot fit evenly around a vertex.
  4. 4
    Step 4: Therefore, regular pentagons cannot tile the plane.

Answer

No — regular pentagons cannot tile the plane because 108° does not divide 360° evenly.
For regular polygons to tile the plane, their interior angle must divide 360° exactly. Only equilateral triangles (60°), squares (90°), and regular hexagons (120°) satisfy this among regular polygons, as 360/60=6, 360/90=4, 360/120=3.

Example 2

medium
A kitchen floor 300\,\text{cm}\times 240\,\text{cm} is tiled with 20\,\text{cm}\times 20\,\text{cm} square tiles. How many tiles are needed? If tiles cost \$2.50 each, what is the total cost?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which of these regular polygons can tile the plane on their own: equilateral triangle, regular hexagon, regular octagon? Explain using interior angles.

Example 2

medium
A wall 180\,\text{cm}\times 120\,\text{cm} is covered with rectangular tiles 15\,\text{cm}\times 10\,\text{cm}. How many tiles are needed? Verify using areas.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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