Tiling Intuition Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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: (5−2)×180°5=540°5=108°\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°360°.
  3. 3
    Step 3: 360°÷108°=3.33...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°108° does not divide 360°360° evenly.
For regular polygons to tile the plane, their interior angle must divide 360°360° exactly. Only equilateral triangles (60°60°), squares (90°90°), and regular hexagons (120°120°) satisfy this among regular polygons, as 360/60=6360/60=6, 360/90=4360/90=4, 360/120=3360/120=3.

About Tiling Intuition

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

Learn more about Tiling Intuition →

More Tiling Intuition Examples

Example 2 medium

A kitchen floor [formula] is tiled with [formula] square tiles. How many tiles are needed? If tiles

Example 3 easy

Which of these regular polygons can tile the plane on their own: equilateral triangle, regular hexag

Example 4 medium

A wall [formula] is covered with rectangular tiles [formula]. How many tiles are needed? Verify usin

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