Which shapes can cover a floor?
With identical tiles, which shapes can cover a floor without leaving gaps?
Explore
The dot is the point on the floor. Each tile shows the angle of the corner that touches it. gap overlap
Test the five shapes
The tiles are regular: equal sides and equal corners. Pick a shape below the drawing. Before the first tile, bet how many of them close the circle around the point, with no gap and no overlap. Then tap the shape once for each tile.
If you think there will always be a gap or an overlap, tap Never closes.
| Shape | Around the point | Covers on its own? |
|---|
Mix two shapes
Mixing different shapes at the same point opens up new ways to cover the floor. Using at least two shapes, close the circle at 360° in a way that covers the whole floor.
Tiles go in the order you tap them, turning around the point. This floor is separate from the floor in task 1.
A 12-sided polygon
A regular 12-sided polygon has corners of . Two of them plus one tile from the list close the circle around a point. Which tile?
What's going on
Around a point on the floor, the corners of the tiles have to add up to exactly 360°, one full turn. If they add up to less, there's a gap. If they add up to more, one tile ends up on top of another.
The rest of this explanation unlocks when you finish task 1, so it doesn't spoil your guesses.
The angle at a corner of a polygon, measured on the inside, is the interior angle. The interior angles of the triangle (60°), the square (90°), and the hexagon (120°) fit a whole number of times into 360°: 6, 4, and 3 times. The regular pentagon's (108°) doesn't: 3 pentagons make 324° and leave a 36° gap, and a fourth one goes over. The octagon (135°) has the same problem. That's why only regular triangles, squares, and hexagons cover the floor on their own.
That holds for regular pentagons. A pentagon that isn't regular can cover the floor: the house in the drawing has corners of 90°, 90°, 135°, 90°, and 135°, and at each point the corners add up to 90° + 90° + 90° + 90° or 135° + 135° + 90°, always 360°.
Covering a floor with tiles, with no gaps and no tile on top of another, makes a tessellation (also called a tiling). Square ceramic floor tiles, hexagonal paving stones on sidewalks and driveways, and a bees' honeycomb are all tessellations.
Where do these angles come from?
Pick a corner of the pentagon and connect it to all the other corners. The pentagon is now split into 3 triangles. The angles of each triangle add up to 180°, so the 5 corners of the pentagon add up to 3 × 180° = 540°. Since the corners are equal, each one is 540° ÷ 5 = 108°.
A polygon with sides always splits into triangles this way. So the angle at each corner is:
| Shape | Triangles | Sum | Each corner |
|---|---|---|---|
| Triangle | 1 | 180° | 60° |
| Square | 2 | 360° | 90° |
| Pentagon | 3 | 540° | 108° |
| Hexagon | 4 | 720° | 120° |
| Octagon | 6 | 1,080° | 135° |
For teachers
Objective
By the end, students can tell whether a regular polygon, or a combination of them, covers a floor on its own, by checking whether the interior angles that meet at a point add up to exactly 360°. It's the same idea as dividing a number by one of its divisors: no remainder, no gap.
Prerequisites
Angles in degrees and the 360° full turn, multiplication and division with remainders, and the names of the polygons. The interior angle formula isn't a prerequisite: the activity shows the angle of each tile and explains the formula afterward, in an optional section.
How to use in class
Plan for 20 to 25 minutes. Project the activity and read the question at the top. Before students start, ask how many pentagons close the circle and count how many say 3 or 4. Let the class test the five shapes in pairs (10 minutes): the activity asks for a guess for each shape before the first tile. Walk around asking how much is left after three pentagons. Bring the class back together and fill in a table on the board with "shape, angle, how many fit, how much is left over," comparing it with the division 360 ÷ angle. Read the explanation together. Finish with the two-shape challenge (8 minutes) and have each pair write down the sum that closed the circle. For high school classes, use the "Where do these angles come from?" section and the 12-sided polygon question.
Common mistakes
Thinking every regular polygon covers the floor, because the square and the hexagon do: it shows up in the guesses, when a student bets a number for the pentagon and the octagon. Thinking no pentagon covers the floor: the explanation shows the house, a pentagon that isn't regular and does cover it. Thinking "almost closes" is good enough: the student stops at three pentagons and says they cover the floor. Ask which tile fits in a 36° gap. Thinking more sides is always worse (the hexagon works and the pentagon doesn't): ask them to compare the angles in the table instead of the sides. In the challenge, thinking that adding up to 360° at one point is enough: triangle, triangle, hexagon, hexagon, in that order, closes the circle, but you can't repeat that same order at every point on the floor. The activity marks the nearby point where the tiles don't fit and says how much is missing there. The order triangle, hexagon, triangle, hexagon works.
Discussion questions
- Why do 3 pentagons leave a gap when 3 hexagons close the circle?
- Where have you seen a floor or a wall covered in hexagons? What about octagons with squares?
- Why do bees build honeycombs out of hexagons instead of triangles or squares, if all three cover the plane?
- The house covers the floor and the regular pentagon doesn't. What's different about their corners?
- Why can't any regular polygon with more than 6 sides cover the floor on its own?
Based on Contemporary Mathematics, OpenStax, section 10.5, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.