Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7, Unit 4, lesson 4, angles in a triangle. The angles of a triangle add to what total?
Answer
180 degrees. - From Year 7, Unit 4, lesson 2, angles at a point. Angles round a point add to what total?
Answer
360 degrees. - From Lesson 5. Does the order of a reflection and a translation always change the result?
Answer
Not always. It can change where the shape ends up, but when the translation has no sideways move the two orders agree.
Words used in this lesson
- tessellation
- a pattern of shapes that covers a flat surface with no gaps and no overlaps. For example: a tiled floor.
- vertex
- a corner where edges meet. For example: the corner of a tile.
- interior angle
- the angle inside a polygon at a corner. For example: 120° in a regular hexagon.
- regular polygon
- a polygon with all sides and all angles equal. For example: a square.
- semi-regular
- two or more regular polygons with the same arrangement at every vertex. For example: 4.8.8.
Tiling a courtyard floor
A designer in Marrakesh wants a courtyard floor made only of identical regular pentagons, laid edge to edge with no gaps. Each corner of a regular pentagon is 108 degrees. At a point where the corners meet, the angles must fill a full turn.
Predict first: how many pentagons can meet at one point, and will they fit exactly? Then use the slider to see the pentagon and the other regular polygons.
n = 8: each corner is 135°. 360 ÷ 135 ≈ 2.7, so 2 copies fit and leave a gap of 90°. It cannot tile on its own.
Show me the steps
- n = 8: each corner is 135°, and 360 ÷ 135 is about 2.7. Two octagons use 270° and leave a gap of 90°, so octagons do not tile on their own.
- n = 3: each corner is 60°, and 360 ÷ 60 = 6. Six triangles fill the point exactly, so triangles tile.
- n = 4: 90°, and 360 ÷ 90 = 4. n = 6: 120°, and 360 ÷ 120 = 3. Squares and hexagons fill the point exactly, so they tile.
- n = 5: 108°, and 3 copies use 324°, leaving 36°. n = 12: 150°, and only 2 copies fit, leaving 60°. Neither tiles on its own.
A tessellation covers a plane with no gaps and no overlaps, so the angles that meet at each vertex add to exactly 360°.
A regular polygon with n sides has each interior angle (n − 2) × 180° ÷ n. On its own it can tile only if 360 ÷ that angle is a whole number: this happens only for the triangle (60°), the square (90°) and the hexagon (120°). These give the three regular tessellations.
A semi-regular tessellation uses two or more kinds of regular polygon with the same arrangement at every vertex, for example 4.8.8 (a square and two octagons).
Mixing polygons
Each number in a name such as 4.8.8 is the number of sides of one polygon round the vertex. Step through the arrangements and read the angle sum.
4.8.8 is a square and two octagons: 90 + 135 + 135 = 360. The angles fill the point exactly.
Show me the steps
- 4.8.8: a square and two octagons. 90 + 135 + 135 = 360, so the angles fill the point exactly.
- 3.6.3.6: triangle, hexagon, triangle, hexagon. 60 + 120 + 60 + 120 = 360, so it fills the point.
- 3.3.3.4.4: three triangles and two squares. 60 + 60 + 60 + 90 + 90 = 360, so it fills the point.
- 3.4.5: 60 + 90 + 108 = 258, leaving a gap of 102°. 5.5.5: 108 × 3 = 324, leaving 36°. These do not fit.
The sides must match as well as the angles. The polygons in these arrangements all have the same side length, so the edges meet exactly. And a gap that is small is still a gap: three pentagons leave only 36°, which is easy to miss on a drawing, but the floor would not close.
Worked examples
Can regular decagons (10 sides) tessellate on their own?
- Interior angle: (10 − 2) × 180 ÷ 10 = 1440 ÷ 10 = 144°.
- 360 ÷ 144 = 2.5, which is not a whole number.
- Two decagons use 288° and leave 72°, and a third would overlap. They cannot.
Show that a triangle and two regular dodecagons can meet at a vertex. The interior angle of a dodecagon is (12 − 2) × 180 ÷ 12 = 150°. Then 60 + 150 + 150 = 360, so the arrangement 3.12.12 fills the point.
A regular polygon has an interior angle of 156°. How many sides has it?
- The exterior angle is 180 − 156 = 24°.
- The exterior angles of any polygon add to 360°, so n = 360 ÷ 24 = 15.
Squares (90°) and equilateral triangles (60°) meet at a vertex. Which mixes fill 360° exactly? Try each number of squares: 0 squares need 360 ÷ 60 = 6 triangles; 1 square leaves 270, which is 4.5 triangles (no); 2 squares leave 180, which is 3 triangles; 3 squares leave 90, which is 1.5 triangles (no); 4 squares need no triangles. The mixed one is 3.3.3.4.4 (or 3.3.4.3.4).
Your go. The first step is given. Can regular nonagons (9 sides) tessellate on their own?
- Interior angle: (9 − 2) × 180 ÷ 9 = 140°.
- Divide 360 by the angle and decide.
Check the answer
360 ÷ 140 is about 2.6, not a whole number, so nonagons cannot tessellate on their own.
Check before you go on
1. Which regular polygon can tessellate on its own?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A regular polygon has 18 sides. What is each interior angle, in degrees?
3. A regular pentagon and a regular hexagon meet at a vertex of a tiling. What angle, in degrees, is left for the rest of the vertex?
Level 2 · Building
4. A regular polygon has an exterior angle of 72°. How many sides has it?
5. How many whole regular 20-sided polygons can meet round a point without overlapping? (Their sides are equal and they meet edge to edge.)
6. Which group of regular polygons fits exactly round a point?
Level 3 · Stretch
7. Two equilateral triangles, a square and a fourth regular polygon meet at a vertex and fill it exactly. How many sides has the fourth polygon?
8. Two regular hexagons and some equilateral triangles meet at a vertex and fill it exactly. How many triangles are there?
Something to think about
In the first figure the regular pentagon leaves a gap of 36 degrees. A designer says: “Then I will just make the pentagons a little bigger so they close the gap.” Will that work? Say why or why not.
One way to think about it
No. The gap is an angle, so it does not depend on the size of the pentagons. A regular pentagon's corners are always 108 degrees, however big it is, so three copies always leave 36 degrees.
Exit ticket
9. Which statement correctly explains a tessellation?