Extra: tessellations
Fit copies of a regular polygon round a point and see which fit with no gaps and no overlaps, why the regular pentagon leaves a gap, and tile with two kinds of polygon.
- France: grade 11
- Iran: grade 8
Guide
This is an extra studio (Iran's Year 8 enrichment book, "Polygons", lesson «کاشیکاری»). It is not on the KS3 programme of study, but it uses KS3 angle facts.
Covering the plane with polygons, with no gaps and no overlaps, is a tessellation. When the polygons meet edge to edge, the angles round every vertex must add up to exactly 360°.
Each interior angle is 60° in an equilateral triangle, 90° in a square, 108° in a regular pentagon, 120° in a regular hexagon and 135° in a regular octagon. A regular polygon tessellates on its own only if its angle divides 360: 6 triangles, 4 squares, 3 hexagons. The regular pentagon does not.
Two kinds of polygon can work too: two regular octagons and a square (135 + 135 + 90 = 360), or three triangles and two squares. Geometric tile patterns are a shared heritage of many cultures.