Tessellation Math Example 4

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

medium

A tiling uses regular octagons and squares. One proposed vertex arrangement has

1

octagon and

2

squares. Does this satisfy the

360°

vertex condition? What arrangement actually works?

1
Interior angle of regular octagon: $\dfrac{(8-2)\times180°}{8} = 135°$ .
2
Interior angle of square: $90°$ .
3
Proposed $1$ octagon $+ 2$ squares: $135° + 90° + 90° = 315° \neq 360°$ . ✗
4
Valid arrangement $4.8.8$ ( $1$ square $+ 2$ octagons): $90° + 135° + 135° = 360°$ . ✓

Answer

One octagon

+

two squares gives

315°

and fails; the valid tiling is

4.8.8

(one square

+

two octagons), giving

360°

Even a small error in the polygon count (swapping how many octagons and squares appear at a vertex) breaks the

360°

condition. Systematically computing and checking the angle sum is the reliable method for verifying any proposed tessellation.

About Tessellation

A tessellation is a pattern that covers an infinite plane with repeated geometric shapes, leaving no gaps and having no overlaps.

Explain why regular hexagons tessellate the plane but regular pentagons do not.

A proposed semi-regular tiling places [formula] triangles and [formula] squares at every vertex. Ver

Does an equilateral triangle tessellate the plane? Justify your answer using the interior angle.