Polygons · Ages 13–14
Tessellations
The angles at each vertex must make exactly 360°.
How to do it
Method
The angles at each vertex must make exactly 360°
Hexagons: 3 × 120° = 360°, yes; pentagons: 360° ÷ 108° isn't whole, no
Another way
Add the angles
135° + 135° + 90° = 360°: two octagons and a square
In full
Repeating one shape (or a few) to cover a whole plane, with no gaps and no overlaps, is a tessellation. The key is at each vertex: the angles meeting at a point must add up to exactly 360°. With one regular polygon: equilateral triangles, 6 × 60° = 360°; squares, 4 × 90° = 360°; regular hexagons, 3 × 120° = 360°. All of them work. A regular pentagon's angles are 108°: three make 324°, leaving a gap of 36°, and four are more than 360°, so it can't. One way: check whether 360° ÷ the interior angle is a whole number. Another: add the angles and see whether they make exactly 360°. Several regular polygons together can work too: two regular octagons and a square, 135° + 135° + 90° = 360°. Any triangle and any quadrilateral tessellates: a quadrilateral's four angles make exactly 360°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Regular 8-sided polygons alone can tessellate a plane.
True or false?
Show the answer and working
Answer: False
Each angle of a regular 8-sided polygon is 135°. At every vertex the angles must make exactly 360°, so 360° ÷ the angle must be whole: only triangles, squares and hexagons manage it.
Example 2
Any quadrilateral tessellates: at every vertex its four angles meet, one of each. A quadrilateral's angles are 91°, 112°, 95° and x. x =
Show the answer and working
Answer: 62°
A quadrilateral's angles add to 360°, a full turn at the point: x = 360° − 298° = 62°.
Example 3
20 congruent isosceles triangles of card have their apex angles (between the two equal sides) put together at one point, fitting exactly round it with no gap and no overlap. Each apex angle = °, and each base angle = °.
Fill in each blank
Show the answer and working
Answer: 18, 81
The 20 apex angles round the point make 360°: each is 360° ÷ 20 = 18°. The two equal base angles make 180° − 18° = 162° (∠ sum of △), so each is 162° ÷ 2 = 81°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A floor is to be covered with one kind of regular polygon tile, each side 33 cm. Which kind covers it with no gaps and no overlaps?
- A
- B
- C
- D
Choose the right picture
Answer:
2.6 regular 3-sided polygons placed round a point fit exactly, with no gap and no overlap.
True or false?
Answer:
3.Any quadrilateral tessellates: at every vertex its four angles meet, one of each. A quadrilateral's angles are 103°, 70°, 63° and x. x =
Answer:
4.6 congruent isosceles triangles of card have their apex angles (between the two equal sides) put together at one point, fitting exactly round it with no gap and no overlap. Each apex angle = °, and each base angle = °.
Fill in each blank
Answer:
5.A regular 6-sided polygon and a regular 6-sided polygon meet at a point. They are ° short of a full turn, and one regular polygon with sides fills the gap exactly.
Fill in each blank
Answer:
All the answers
- 1. C
- 2. True
- 3. 124°
- 4. 60, 60
- 5. 120, 6
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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