Geometry and proof · Ages 13–14

Tessellations worksheet

Part of: Polygons

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Geometry and proof · Ages 13–14

Numsprite
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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Three regular polygons meet at a point, leaving a gap. x =
Answer:
2.Any quadrilateral tessellates: at every vertex its four angles meet, one of each. A quadrilateral's angles are 72°, 99°, 74° and x. x =
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3.Any quadrilateral tessellates: at every vertex its four angles meet, one of each. A quadrilateral's angles are 109°, 75°, 74° and x. x =
Answer:
4.A regular 4-sided polygon and a regular 5-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
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5.9 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
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6.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
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7.Regular 8-sided polygons alone can tessellate a plane.
True or false?
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8.A floor is to be covered with one kind of regular polygon tile, each side 21 cm. Which kind covers it with no gaps and no overlaps?
- A
- B
- C
- D
Choose the right picture
Answer:
9.A floor is to be covered with one kind of regular polygon tile, each side 37 cm. Which kind covers it with no gaps and no overlaps?
- A
- B
- C
- D
Choose the right picture
Answer:
10.3 regular 5-sided polygons placed round a point fit exactly, with no gap and no overlap.
True or false?
Answer:
11.Regular polygon cards of side 7 cm are put together round a point. In which picture do they fit exactly round it, with no gap?
- A
- B
- C
- D
Choose the right picture
Answer:
12.Regular polygon cards of side 5 cm are put together round a point. In which picture do they fit exactly round it, with no gap?
- A
- B
- C
- D
Choose the right picture
Answer:

13.A floor is tiled with equilateral triangles and squares. At one point, 3 triangles and 1 squares already meet. How many degrees are still free there?
Working:
Answer:
14.2 regular 8-sided tiles and 1 square tiles placed round one point fit exactly, with no gap.
True or false?
Working:
Answer:
15.Why can't regular pentagon tiles alone cover a floor?
- A. Each angle is 108°: three make 324°, short of 360°, and four make 432°, too many
- B. A pentagon has an odd number of sides
- C. Its angle sum, 540°, isn't a multiple of 360°
- D. A regular pentagon has no line of symmetry
Choose the right answer
Working:
Answer:
16.A floor is tessellated with one congruent triangular tile; two of its angles are 38° and 69°. At every vertex each of its three angles appears twice, making a full turn. The third angle =
Working:
Answer:
17.Two regular polygons meet at a point, leaving a gap x. x = °, and one regular polygon with sides fills it exactly.
Fill in each blank
Working:
Answer:
18.A floor is tessellated with one congruent triangular tile; two of its angles are 89° and 30°. At every vertex each of its three angles appears twice, making a full turn. The third angle =
Working:
Answer:

Show the answers and working

Tessellations worksheet: Answers and working

1. 22°
Their angles are 90°, 108° and 140°, 338° together; angles round a point make 360°: x = 360° − 338° = 22°.
2. 115°
A quadrilateral's angles add to 360°, a full turn at the point: x = 360° − 245° = 115°.
3. 102°
A quadrilateral's angles add to 360°, a full turn at the point: x = 360° − 258° = 102°.
4. 162, 20
90° + 108° leaves 360° − 90° − 108° = 162°. A regular polygon with 162° angles has exterior angles of 180° − 162° = 18°, so 360° ÷ 18° = 20 sides.
5. 40, 70
The 9 apex angles round the point make 360°: each is 360° ÷ 9 = 40°. The two equal base angles make 180° − 40° = 140° (∠ sum of △), so each is 140° ÷ 2 = 70°.
6. 120, 6
120° + 120° leaves 360° − 120° − 120° = 120°. A regular polygon with 120° angles has exterior angles of 180° − 120° = 60°, so 360° ÷ 60° = 6 sides.

7. 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.
8. C
Only triangles (6 × 60°), squares (4 × 90°) and hexagons (3 × 120°) have angles that make exactly 360° at a point.
9. B
Only triangles (6 × 60°), squares (4 × 90°) and hexagons (3 × 120°) have angles that make exactly 360° at a point.
10. False
Each angle is 108°, so 3 of them make 3 × 108° = 324°; it must be exactly 360°.
11. B
The angles: triangle 60°, square 90°, hexagon 120°, octagon 135°, 12-sided 150°. Only angles adding to exactly 360° leave no gap.
12. C
The angles: triangle 60°, square 90°, hexagon 120°, octagon 135°, 12-sided 150°. Only angles adding to exactly 360° leave no gap.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 90°
A triangle's angle is 60° and a square's 90°: 360° − 3 × 60° − 1 × 90° = 90°.
14. True
Add the angles at the point: 360° in all, and it must be exactly 360° to leave no gap.
15. A. Each angle is 108°: three make 324°, short of 360°, and four make 432°, too many
At every point the angles must make exactly 360°; 360° ÷ 108° isn't whole, so pentagons can't.
16. 73°
2 × (38° + 69° + the third) = 360°, so the three add to 180°: the third = 180° − 38° − 69° = 73°.
17. 135, 8
x = 360° less the two angles = 135°. A regular polygon with 135° angles has exterior angles of 180° − 135°, so 360° ÷ (180° − 135°) = 8 sides.
18. 61°
2 × (89° + 30° + the third) = 360°, so the three add to 180°: the third = 180° − 89° − 30° = 61°.

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