2. Can every shape tile a floor?

Patterns Everywhere overview Lesson 2 of 6 · Can every shape tile a floor?

Some shapes fit together neatly. Others leave mysterious little gaps.

In this lesson, you are going to experiment first. Your job is to find out which shapes can cover a surface without gaps and without overlaps.

You can slide, flip and turn the shapes.

💡 New to degrees? Open this.

A turn means rotating a shape around a point. The picture below shows some clockwise turns.

Examples of clockwise turns showing 0, 30, 60 and 90 degree rotations of a triangle and square

90° is a quarter turn. 360° is one complete turn back to where you started.

Start with a prediction

Try it

Which of these shapes do you think could cover a floor completely?

■▲⬢●⬟

Make a guess before you test them. It is completely fine if your guess changes later.

Tiling lab

Use the interactive board to place copies of a shape, move them around and turn them. Try to make the pieces meet exactly.

Interactive lab

Build a tiling

Add a shape, drag it into place, then turn or flip it. Every new piece starts in the visible board.

Choose a shape to begin. Drag pieces freely into place. Use Turn to change their orientation.

Try the shapes one at a time. Before deciding that a shape does not work, try turning it.

What did you notice?

Tap a card to reveal the answer. Compare what you discovered in the Tiling Lab with the explanation.

Squares Can you make every edge meet another edge? Tap to reveal ↻Tap to close ↺
Yes — squares tessellate. Four 90° corners meet to make a complete 360° turn, so there are no gaps.
Triangles What happens when you turn alternate triangles? Tap to reveal ↻Tap to close ↺
Yes — equilateral triangles tessellate. Turning alternate triangles makes the edges fit perfectly. Six 60° corners make 360°.
Hexagons Do they remind you of anything you have seen in nature? Tap to reveal ↻Tap to close ↺
Yes — regular hexagons tessellate. Three 120° corners meet to make 360°, like the cells in a honeycomb.
Circles Push them as close together as you can. What is left between them? Tap to reveal ↻Tap to close ↺
No — circles do not tessellate. Curved edges leave small gaps between neighbouring circles.
Regular pentagons Can their corners fit exactly around one point? Tap to reveal ↻Tap to close ↺
three regular pentagons 108° 108° 108° 36° gap angle view
3 × 108° = 324° < 360°
No — regular pentagons do not tessellate. Three 108° corners make 324°, leaving a 36° gap. Four corners would total 432° and overlap, so no whole number of regular-pentagon corners can fit exactly around a point.

A new mathematical word

Definition
Tessellation

When copies of shapes cover a surface without gaps and without overlaps, the pattern is called a tessellation.

Squares tessellate. Triangles tessellate. Regular hexagons tessellate. Circles do not tessellate, because gaps remain between them. Regular pentagons do not tessellate either.

Why do some shapes fit?

Look at a point where several corners meet. If the corners fit perfectly around that point, they make one complete turn.

▲ ▲ ▲ ▲ ▲ ▲

Triangles: 6 corners of 60° make 360°.

■ ■ ■ ■

Squares: 4 corners of 90° make 360°.

⬢ ⬢ ⬢

Regular hexagons: 3 corners of 120° make 360°.

Go further

The angle test gives us a quick way to investigate regular shapes.

For a regular pentagon, each corner is 108°. Three corners make 324°, which leaves a gap. Four make 432°, which is too much.

Try this idea with another regular polygon: find one corner angle and ask whether a whole number of those angles can make exactly 360°.

Practice puzzle

You try to fit copies of a shape together and your first arrangement leaves a gap. Can you conclude that the shape cannot tessellate?

Reveal answer
No. One unsuccessful arrangement is not enough. Move or turn the shapes and try again. To show that a shape cannot tessellate, you need a reason that rules out every possible arrangement — for example, the angle argument for regular pentagons.
Printable resources

Lesson 2: Can every shape tile a floor?

Download the Lesson 2 printable resources to explore tessellations, gaps, overlaps and how shapes can fit together to cover a surface.

Get the Lesson 2 resources →

Next we will investigate another kind of pattern: symmetry. Instead of asking whether shapes fit together, we will ask whether one part of a picture matches another.

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Patterns Everywhere overview Lesson 2 of 6 · Can every shape tile a floor?