Tessellation Activity for Kids: Which Shapes Tile a Floor

Hand a child eight paper pentagons and ask them to cover a tray with no gaps. They will try for about four minutes, get three pentagons to meet, and find a wedge of empty space they cannot fill no matter how they shuffle. That wedge is 36 degrees wide, and it is the whole lesson. This tessellation activity for kids works because the failure is visible and measurable. Then you hand them eight octagons, which also fail, and the leftover gap turns out to be exactly a square. That is a real floor pattern they have walked on.

Key Takeaways

• You’ll cut identical card shapes and try to tile a tray with each one, then measure the gaps that are left over.
• It teaches that interior angles have to add to 360° around a point, which is why only three regular shapes tile a plane alone.
• Best from 6 for the tiling, from 9 for the angle arithmetic. The cutting is the slow part, so an adult should pre-cut most of it.
• The reveal is the octagon: its gaps are perfect squares, which is a tiling you have seen in real bathrooms.
• The best-known “fact” here is wrong. Regular pentagons cannot tile, but 15 families of irregular pentagons can.

What You’ll Need

Item	How much	Where to get it	If you don’t have it
Card or thick paper	4 sheets	Stationery drawer	Cereal-box card; it cuts cleanly and stacks
Scissors	1 adult pair, 1 child pair	Kitchen drawer	Adult cuts everything in advance
Protractor	1	School supplies	Skip the measuring and compare gaps by eye
Ruler and pencil	1 each	Anywhere	Any straight edge
A tray or baking sheet	1	Kitchen	A sheet of A3 paper taped to the table
Sticky tape	1 roll	Drawer	Glue stick
Paper clips	4	Desk	Hold card layers while cutting multiples

Time: prep 25 min of cutting, activity 35 min, cleanup 5 min Cost: everything is probably in the house Mess level: low, but there will be a lot of card offcuts

Do the Experiment: A Tessellation Activity for Kids

Step 1 – Make one master template of each shape

Draw five shapes on card, each with 5 cm sides: an equilateral triangle, a square, a regular pentagon, a regular hexagon and a regular octagon. Use a protractor for the pentagon and octagon, because freehand ones will not tile even if the shape would, and you will spend the afternoon debugging your own cutting.

Step 2 – Cut eight copies of each, four layers at a time

Stack four pieces of card, clip them together, trace the template and cut through all four. You need roughly eight triangles, eight squares, eight pentagons, eight hexagons and eight octagons. This is 25 minutes of work and it is the only tedious part, so do it the night before.

Step 3 – Tile the easy ones first

Give them the squares. Then the triangles. Both fill the tray with no gaps, and the triangles will surprise them: you have to flip every other one upside down. Let them find that themselves.

Step 4 – Try the pentagons, then measure the gap

Pentagons will go three-around-a-point and leave a wedge. Have your kid slide them around for a few minutes before you say anything. Then put a protractor on the gap. It is 36°. Write that number down.

Step 5 – Try the hexagons, then the octagons

Hexagons work, three meeting at every point. Octagons do not: two of them leave a gap. Measure it. The gap is 90°, a perfect right angle. Now give them the squares back and let them fill the octagon gaps.

Step 6 – Build the angle table

Make a table with the shape, the number of sides, the interior angle and 360 divided by that angle. Triangle 60 and 6. Square 90 and 4. Pentagon 108 and 3.33. Hexagon 120 and 3. Octagon 135 and 2.67. Ask which rows have a whole number in the last column.

Step 7 – Invent a tile that works

Take a square. Cut a curvy piece off the left edge and tape it onto the right edge exactly as it was, without rotating it. Cut a bump off the top and tape it to the bottom the same way. The new shape looks nothing like a square and tiles perfectly, because you did not change its area or how its edges match. Make eight and prove it. Then draw a face on each.

What you should see: Squares, triangles and hexagons tile with no gaps at all. Pentagons leave a repeating 36° wedge, which is the clearest result in the activity. Octagons leave 90° square holes, and filling them with your squares gives the octagon-and-square pattern found in tiled floors everywhere. The invented tile from step 7 tiles on the first attempt if you taped the pieces on without rotating them, and fails visibly if you rotated one.

What’s Actually Going On

A tessellation is a covering of a flat surface by shapes with no gaps and no overlaps. The constraint that decides everything is local: at any point where corners meet, the angles must add to exactly 360°, because that is how much turning there is around a point.

The interior angle of a regular polygon with n sides is (n − 2) × 180 / n. Plug in the numbers and you get 60° for a triangle, 90° for a square, 108° for a pentagon, 120° for a hexagon, 135° for an octagon. For copies of a single regular polygon to meet at a point, 360 has to be a whole-number multiple of that angle. Sixty goes into 360 six times. Ninety goes four times. One hundred and twenty goes three times. One hundred and eight does not go at all, and neither does 135. That is the complete argument, and it is why there are exactly three regular tilings of the plane and no more. Your kid’s protractor measurement of 36° is 360 − 3 × 108, which is the leftover after the third pentagon.

Allow more than one kind of regular polygon and the count goes up: there are eight semi-regular, or Archimedean, tilings, in which every vertex looks the same but more than one polygon is used. The octagon-and-square floor is one of them.

Now drop the requirement that the shapes be regular, and the popular summary of this topic collapses in a useful way. Every triangle tiles the plane, not just the equilateral one: rotate any triangle 180° about the midpoint of a side and you get a parallelogram, which tiles. Every quadrilateral tiles too, including bent, non-convex ones, by the same midpoint-rotation trick. So “only regular shapes tile” is wrong, and it is worth saying out loud, because it is the thing most children take away from a worksheet.

Pentagons are where this gets genuinely hard, and the history is recent enough to be startling. Regular pentagons cannot tile, but irregular convex pentagons can, and classifying them took 97 years. The fourteenth type was found in 1985. The fifteenth was found in 2015 by Casey Mann, Jennifer McLoud-Mann and David Von Derau using a computer search. In 2017 Michaël Rao published an exhaustive computer search concluding that there are no more than those 15 families. A twelve-year-old can understand that statement, which is a rare thing in modern mathematics.

One more, even more recent. In 2023 a retired printing technician named David Smith found a single tile, nicknamed the hat, that tiles the plane but never periodically: the pattern never repeats by sliding. Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss published the proof in Combinatorial Theory, settling a question that had been open for decades. Whether a single aperiodic tile existed was genuinely unknown when your child’s teacher was at school.

How to Explain It to Your Kid

Ages 5–8: no gaps allowed

Use the word “gaps” and nothing else. Give them squares, then triangles, then pentagons, in that order, and let the pentagons annoy them. Then ask the question that does the work: “Can you push them closer together?” They will try. They cannot. Say: “Some shapes fit all the way round a corner and some don’t, and you just found one that doesn’t.” Then do the octagons and let them discover that squares rescue it. That rescue is the part they will remember.

Ages 9–12: 360 around a point

Teach the angle sum at a point first, by fitting six triangles around one dot and counting 6 × 60. Then hand them the protractor and the table from step 6. The division in the last column is the whole idea, and they can do it. Then set the test: predict, from the arithmetic alone, whether a regular 12-sided shape tiles. Its interior angle is 150°, and 360 ÷ 150 is 2.4, so it does not. They can answer that without cutting a single dodecagon, which is the moment arithmetic starts feeling useful.

Ages 13+: the formula, and the open questions

Derive (n − 2) × 180 / n with them rather than giving it, then have them solve for which n make 360 divisible by it. There are three solutions and they can prove there are only three. Then set the investigation: take a random scalene triangle and a random bent quadrilateral, and show by construction that each tiles. After that, tell them about the 15 pentagon families and the 2023 hat tile, and that parts of this were settled in their lifetime.

The question to ask: “I have a shape with an interior angle of 72° at every corner. Without cutting anything out, will copies of it fit around a point with no gaps? What about 100°?”

Why Teaching This Early Pays Off

Tiling is where angle arithmetic stops being a worksheet and becomes a tool that answers a question you actually care about. A child who has measured a 36° gap with a protractor has a reason to believe that angles add up, and that is the foundation for everything from triangle geometry to trigonometry.

There is a second payoff that matters more. This is one of very few school topics where a child can be told, truthfully, that part of it was unknown until recently and part was decided by a computer search. The fifteenth pentagon in 2015, Rao’s exhaustive search in 2017, the hat tile in 2023. Mathematics presented as a finished body of facts is a lie that costs us students; tiling is a cheap way to tell the truth.

The research caution applies here too. Alfieri and colleagues found that discovery works when it is guided, and fails when children are simply left to explore. The pentagon failure is not self-explanatory: a child shuffling pentagons for ten minutes without the protractor and the arithmetic learns only that pentagons are annoying. You have to supply the 360. Do that, and the Framework for K-12 Science Education practice of using mathematics and computational thinking is sitting right there on your kitchen table.

Before You Start: What Parents Should Know

Best age	6 for the tiling, 9 for the angles. A six-year-old gets everything out of steps 3 to 5 and nothing out of step 6, which is fine.
Supervision	Adult does the cutting, especially the pentagons and octagons. Inaccurate shapes produce a wrong result that looks like a real result.
Real hazards	Scissors. Card offcuts are a choking risk for under-3s, so sweep them into a bag rather than leaving them on the floor.
Setup	A clear table, plus a tray to keep the growing pattern from sliding apart when someone leans on it.
Frustration point	Step 4. Some children decide they are bad at puzzles rather than that the shape is impossible. Say early that one of these shapes genuinely cannot be done.
Don’t	Don’t cut the pentagons or octagons freehand, and don’t rotate the pieces when you tape them on in step 7. Both failures look like the maths broke.

When It Doesn’t Work

• The hexagons leave small gaps. The shapes are not accurate. Cut from one template rather than tracing each new shape from the previous one, which compounds error fast.
• The triangles will not fit together. They are all pointing the same way. Every other triangle has to be flipped. Say nothing for two minutes, then rotate one yourself.
• The pattern slides apart. Work inside a tray, or put a strip of tape at the starting corner. Fifteen loose card shapes are one sneeze from over.
• The invented tile from step 7 does not tile. A cut piece was rotated or flipped before being taped on. Slide, do not turn. Re-cut and try again; it works every time when the translation is exact.
• The octagon gap does not look square. The octagon is not regular. Check that all eight angles are 135°; a hand-drawn octagon is usually two different shapes at once.

Take It Further

Count the semi-regular tilings yourself. Work out which combinations of regular polygon angles add to exactly 360: 90 + 135 + 135 works, 60 + 60 + 120 + 120 works, and so on. Then try to build each one from your cut shapes. There are eight that produce a valid tiling with every vertex alike, and finding them by arithmetic is a proper afternoon’s work for a teenager.

Make an Escher animal and a full sheet. Use the step 7 method on a hexagon instead of a square, with three slides instead of two. Trace twenty copies onto a large sheet and colour them. This is how Escher’s lizard tilings were actually built, and the method transfers exactly.

Test the quadrilateral claim. Draw a deliberately ugly four-sided shape, one with a dent in it. Cut eight copies. Show that it tiles by rotating each one 180° about the midpoint of a shared edge. This is a proof a twelve-year-old can carry out with scissors.

What to Watch For Over the Next 3 Months

• Week 4: They start noticing real floors and walls, usually in a shop or a bathroom, and announcing which tiling it is.
• Month 2: The phrase “that adds up to 360” appears in a different context, often while doing something unrelated with angles.
• Month 3: The question worth encouraging is why pentagons are special. You can honestly say that nobody fully understood pentagon tilings until 2017, and that the answer came out of a computer search.

Frequently Asked Questions

Why can’t regular pentagons tile a floor?

Because their interior angle is 108°, and 360 is not a whole-number multiple of 108. Three pentagons around a point give 324°, leaving a 36° wedge, and no arrangement closes it. Only 60°, 90° and 120° divide 360 exactly among regular polygon angles.

So pentagons can never tile?

Regular ones cannot. Irregular convex pentagons can, and there are exactly 15 families of them. The fifteenth was discovered in 2015 and the classification was completed by an exhaustive computer search published in 2017.

Do I have to cut all those shapes by hand?

You can print templates, which saves the tracing but not the cutting. Stacking four sheets of card and cutting through all of them with paper clips holding the stack is the fastest honest method, and it also guarantees the copies are identical, which matters more than speed.

What is the point if it is just shapes?

The angle sum around a point is one of the few geometric facts that generalises into almost everything later: triangle angle sums, polygon interiors, rotational symmetry, and the reason a Mercator map of the globe has to distort. Tiling is the cheapest way to make that fact felt rather than memorised.

Is there really a shape that tiles without ever repeating?

Yes, and it was found recently. In 2023 David Smith identified a single tile, the hat, that covers the plane with no gaps while never repeating by translation. The proof, with Myers, Kaplan and Goodman-Strauss, appeared in Combinatorial Theory. Before that, nobody knew whether one tile could do it.

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About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

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Sources

1. Rao, M. (2017). “Exhaustive search of convex pentagons which tile the plane.” arXiv:1708.00274. https://arxiv.org/abs/1708.00274
2. Mann, C., McLoud-Mann, J., & Von Derau, D. (2015). “Convex pentagons that admit i-block transitive tilings.” arXiv:1510.01186. https://arxiv.org/abs/1510.01186
3. Smith, D., Myers, J. S., Kaplan, C. S., & Goodman-Strauss, C. (2024). “An aperiodic monotile.” Combinatorial Theory, 4(1). https://doi.org/10.5070/C64163843
4. NRICH, University of Cambridge. “Tessellating Triangles.” https://nrich.maths.org/problems/tessellating-triangles
5. Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). “Does discovery-based instruction enhance learning?” Journal of Educational Psychology, 103(1), 1–18. https://doi.org/10.1037/a0021017
6. National Research Council. (2012). A Framework for K-12 Science Education. National Academies Press. https://doi.org/10.17226/13165

Related reading: counting Fibonacci spirals on real pinecones, cutting a Mobius strip in half, and colouring a map with only four crayons.