Tiling Intuition Math Example 3

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

Example 3

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

Solution

  1. 1
    Step 1: Equilateral triangle: interior angle =60°= 60°. 360°/60°=6360°/60° = 6 ✓ (tiles the plane).
  2. 2
    Step 2: Regular hexagon: interior angle =120°= 120°. 360°/120°=3360°/120° = 3 ✓ (tiles the plane).
  3. 3
    Step 3: Regular octagon: interior angle =135°= 135°. 360°/135°=2.67360°/135° = 2.67 ✗ (does not tile the plane alone).

Answer

Equilateral triangles and regular hexagons can tile the plane; regular octagons cannot.
A regular polygon tiles the plane if and only if its interior angle divides 360°360° exactly. Triangles (60°) and hexagons (120°) pass this test; octagons (135°) do not, though octagons can tile in combination with squares.

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 1 easy

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

Example 2 medium

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

Example 4 medium

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

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