What is a tessellation?

A tessellation is a pattern of repeating shapes that fit together with no gaps and no overlaps, covering a surface and able to continue forever. It is also called tiling.

Which shapes tessellate on their own?

Squares, equilateral triangles and regular hexagons each tile by themselves. They work because their angles divide evenly into 360°, so copies meet neatly at every corner.

Why won't a regular pentagon or a circle tessellate?

A regular pentagon's angles are 108°, and three of them make only 324° while four make 432°, so they can never add to exactly 360° around a point. Circles leave curved gaps because their edges don't sit flat against each other.

Why do the angles around a point have to add up to 360°?

A full turn around any point is 360°. If the angles meeting at a corner total less, a gap is left; if they total more, the shapes overlap. Only an exact 360° gives a perfect tessellation.

Where can my child see tessellations in real life?

Common examples include honeycomb in a beehive, brick walls, bathroom and pavement tiles, the scales on a fish, and the plates on a turtle's shell.

Tessellations: Shapes That Tile Forever

Math Interactive lesson Free to play

A tessellation is a pattern made by repeating one or more shapes so they fit together perfectly across a surface — with no gaps and no overlaps — and could carry on forever in every direction. The same idea is also called tiling, and you can see it in honeycombs, brick walls, bathroom floors, fish scales and turtle shells.

Tessellations matter because they connect shape, angle and pattern in a way children can see and touch. The key idea is the corner rule: wherever the points of shapes meet, the angles must add up to exactly 360° — a full turn — or a gap or overlap appears. This is why squares, equilateral triangles and regular hexagons tile on their own, while regular pentagons and circles cannot.

Learners come away understanding what makes a shape tessellate, how to test a shape by checking the angles around a meeting point, and how repeating one shape and changing only the colours can create endless original patterns. It builds geometry intuition that later supports area, angles and symmetry.

▶ Play the lesson — free, no signup

What this Spark covers

Shapes that tile forever

🐝🔷🐢 Tessellations: Shapes That Tile Forever A tessellation is a pattern of shapes that fit together perfectly — no gaps and no overlaps — and could go on forever in every direction. Bees build honeycombs out of tessellating hexagons! 🐝 Let's discover which shapes tile forever — and play with making patterns yourself. Tap Next to begin.
No gaps, no overlaps

The two golden rules For shapes to tessellate, they must follow two rules. Tap each tile pattern below to check it. 🟩🟩🟩🟩 Tap me 🔴🔴🔴🔴 Tap me Tap a pattern to reveal its secret… Rule 1 — No gaps: no empty spaces left over. Rule 2 — No overlaps: shapes never sit on top of each other. Circles fail because round edges leave little gaps. Straight edges fit much better!
Which shapes tile?

Tap every shape that tiles forever Some shapes fit together on their own; others leave gaps. Tap the ones you think tessellate. (Hint: think about whether copies fit edge-to-edge.) Square Triangle Hexagon Circle Pentagon Tap the shapes that tile with no gaps. Find all 3! Surprise: a regular pentagon leaves gaps, but squares, triangles and hexagons tile perfectly. 🔺
The corner secret

The corner secret: 360° Where corners meet at a point, the angles must add up to exactly 360° — a full turn — with no space left. Add tiles around the centre dot. Try to reach exactly 360°! ➕ Triangle 60° ↺ Reset Total: 0° Six triangles (6 × 60° = 360°) close up perfectly. That's WHY triangles tessellate!
Fill the wall

Build a tiled wall A real tessellation has no empty spaces. Tap the squares to lay tiles and fill the whole wall — every cell must be covered! 0 of 36 tiles laid. Fill them all! Clear wall Tiles, bathroom floors and brick walls are everyday tessellations. 🧱
Spot them around you

Tessellations are everywhere! Tap each card to reveal a real-life tessellation you can spot. 🐝Honeycomb 🐢Turtle shell 🧱Brick wall 🛁Floor tiles Tap a card… how many can you uncover?
Colour the pattern

Make your own pattern Pick a colour, then tap the triangles. The same shape repeats — but YOUR colours make a brand-new design! Tap triangles to colour them. Colour at least 6! Artist M.C. Escher made amazing tessellation pictures of birds and fish. 🐦🐟
You're a tessellation star!

🌟🔷🌟 You did it! You now understand shapes that tile forever. 🔑 Tessellation = shapes that fit with no gaps and no overlaps, repeating forever. 🟩 Squares, triangles and hexagons tessellate on their own. 🔴 Circles and regular pentagons leave gaps, so they don't. 🎯 The corner secret: angles meeting at a point must add to 360°. 🐝 Look for tessellations in honeycombs, brick walls, floor tiles and turtle shells! Well done, tessellation star! ⭐

Frequently asked questions

What is a tessellation?: A tessellation is a pattern of repeating shapes that fit together with no gaps and no overlaps, covering a surface and able to continue forever. It is also called tiling.
Which shapes tessellate on their own?: Squares, equilateral triangles and regular hexagons each tile by themselves. They work because their angles divide evenly into 360°, so copies meet neatly at every corner.
Why won't a regular pentagon or a circle tessellate?: A regular pentagon's angles are 108°, and three of them make only 324° while four make 432°, so they can never add to exactly 360° around a point. Circles leave curved gaps because their edges don't sit flat against each other.
Why do the angles around a point have to add up to 360°?: A full turn around any point is 360°. If the angles meeting at a corner total less, a gap is left; if they total more, the shapes overlap. Only an exact 360° gives a perfect tessellation.
Where can my child see tessellations in real life?: Common examples include honeycomb in a beehive, brick walls, bathroom and pavement tiles, the scales on a fish, and the plates on a turtle's shell.

Tessellations: Shapes That Tile Forever

What this Spark covers

Frequently asked questions

More Sparks like this