Which shapes can cover a floor with no gaps? Tile it, find out why, solve missions and beat the clock.
Start here. A tessellation (or tiling) covers a flat surface with copies of a shape. There are no gaps and the tiles never overlap. Bathroom floors, honeycombs and brick walls are all tessellations.
Which shapes work? Squares, rectangles, equilateral triangles and regular hexagons all tile a floor by themselves. Circles, regular pentagons and regular octagons do not: they leave gaps.
Why? Look at a corner point. Where tiles meet, their corners must fit around the point and make exactly one full turn, which is 360°. A square corner is 90°, and 4 × 90° = 360°. A triangle corner is 60°, and 6 × 60° = 360°. A hexagon corner is 120°, and 3 × 120° = 360°.
Why pentagons fail. A regular pentagon has corners of 108°. Three of them make 324°, which leaves a 36° gap. Four would make 432°, which is too much, so they overlap. A circle has no corners at all, so it always leaves gaps.
Corner sizes. For a regular shape with some number of sides: (sides − 2) × 180° ÷ sides. That gives triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135° and a 12-sided shape 150°.
Mixing shapes. A regular octagon alone leaves gaps: 2 × 135° = 270° leaves 90°, which is exactly a square corner. So octagons and small squares tile together: 135° + 135° + 90° = 360°. Other mixes that make 360° around a point are 4 triangles + 1 hexagon, 2 triangles + 2 hexagons, and a square + a hexagon + a 12-sided shape.
Any triangle tiles, any four-sided shape tiles. The corners of a triangle add up to 180°. Two copies of the same triangle make a parallelogram, and parallelograms tile. The corners of any four-sided shape add up to 360°, so four copies fit around a point, each corner used once.
Try this above. In Play, tile the floor with each shape in turn and watch what happens. Then pick octagons and add the small squares to fill the holes.
It is a pattern of shapes that covers a flat surface completely, with no gaps and no overlaps. The word comes from "tessera", the Latin word for a small square tile.
360° is one full turn around a point. If the corners add up to less, there is a gap. If they add up to more, the tiles would overlap.
A regular pentagon (all sides and corners equal) cannot. Some pentagons with unequal sides and corners can tile, but they are special shapes. This game uses regular shapes.
Pick the level, not the age. "Just starting" finds which shapes tile and which leave gaps. "I know the basics" uses corner sizes and 360°. "Challenge me" mixes shapes around a point and uses angle sums of triangles and four-sided shapes.
No. It is free and needs no account. Your stars are saved in your own browser.