Geometry and proof · Ages 13–14
Polygons worksheet
Covers: Regular polygons, Angle sum of a polygon, Exterior angles of a polygon, Tessellations, Equilateral triangles, squares and hexagons, Drawing a regular pentagon
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Polygons worksheet
Geometry and proof · Ages 13–14
Numsprite
numsprite.com
Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.The picture shows a regular polygon. It has lines of symmetry.
Answer:
2.The diagonals from A cut the polygon into triangles, so its angles add up to × 180° = °.
Fill in each blank
Answer:
3.In the pentagon, the three angles marked x are equal. x =
Answer:
4.Each exterior angle of a regular polygon is 45°. It has sides.
Answer:
5.One side of the hexagon is extended. x =
Answer:
6.4 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:
7.An arc with centre A and radius AB, and another with centre B and the same radius, cross at C. AB = 4 cm. The perimeter of △ABC =
Answer:
8.A regular hexagon ABCDEF has sides of 4 cm. The diagonal AD =
Answer:
9.Draw a regular pentagon in a circle of radius 7 cm with compasses, a ruler and a set square: put the steps in order.
- A. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn
- B. MN = MC: with centre M and radius MC, draw an arc cutting OA at N
- C. ∠AOC = 90°: draw a circle, centre O and radius 7 cm, and with a set square two diameters AB and CD at right angles
- D. OM = MB: measure OB with the ruler to find its midpoint M
Write the letters in the right order
Answer:
Part B: Choose, or true or false (2 marks each)
10.How many lines of symmetry of a regular 6-sided polygon pass through its vertices?
- A. 12
- B. 3
- C. 6
- D. 4
Choose the right answer
Answer:
11.A floor is to be covered with one kind of regular polygon tile, each side 26 cm. Which kind covers it with no gaps and no overlaps?
- A
- B
- C
- D
Choose the right picture
Answer:
12.Each angle of a regular pentagon is 108°.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A rhombus-shaped tile has four sides of 27 cm, and one of its angles is 68°. Is it a regular polygon?
- A. No: the sides aren't all equal
- B. Yes: all four sides are equal
- C. No: the sides are equal, but not all the angles are
- D. Yes: opposite angles are equal
Choose the right answer
Working:
Answer:
14.Four angles of a five-sided flower bed are 121°, 99°, 118° and 93°. The angles of a pentagon add to °, so the fifth = °.
Fill in each blank
Working:
Answer:
15.A toy car drives once round a track shaped like a regular 5-sided polygon. At each corner it turns °, and back at the start it has turned ° in all.
Fill in each blank
Working:
Answer:
16.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. Its angle sum, 540°, isn't a multiple of 360°
- C. A pentagon has an odd number of sides
- D. A regular pentagon has no line of symmetry
Choose the right answer
Working:
Answer:
17.A regular hexagon ABCDEF is drawn in a circle of radius 7 cm, centre O. The perimeter of △OAB = cm, ∠ABC = °.
Fill in each blank
Working:
Answer:
18.A regular pentagon tile ABCDE is cut along the diagonal AD. ∠ADC =
Working:
Answer:
Show the answers and working
Polygons worksheet: Answers and working
Part A: Working out and filling in
1. 5
A regular polygon has as many lines of symmetry as sides: 5 sides, so 5 lines.
2. 5, 5, 900
From one vertex a 7-sided polygon splits into 7 − 2 = 5 triangles, so its angles add to 5 × 180° = 900°.
3. 117°
3x + 103° + 86° = 540° (∠ sum of polygon), 3x = 351°, x = 117°.
4. 8
Sides = 360° ÷ one exterior angle = 360° ÷ 45° = 8.
5. 53°
The sixth angle = 720° − 593° = 127° (∠ sum of polygon), so x = 180° − 127° = 53° (adj. ∠s on st. line).
6. 90, 45
The 4 apex angles round the point make 360°: each is 360° ÷ 4 = 90°. The two equal base angles make 180° − 90° = 90° (∠ sum of △), so each is 90° ÷ 2 = 45°.
7. 12 cm
AC and BC are both the radius, equal to AB: three sides of 4 cm, perimeter 3 × 4 = 12 cm.
8. 8 cm
AD passes through the centre O: OA and OD are both the radius, equal to the side, 4 cm, so AD = 2 × 4 = 8 cm.
9. C → D → B → A
The diameters at right angles come first, so that OB's midpoint M can be found; M gives the arc and N; CN is the side, marked off and joined last.
Part B: Choose, or true or false
10. B. 3
With an even number of sides, half the lines pass through two opposite vertices and half through the middles of opposite sides: 6 ÷ 2 = 3 pass through vertices.
11. D
Only triangles (6 × 60°), squares (4 × 90°) and hexagons (3 × 120°) have angles that make exactly 360° at a point.
12. True
The angles add to (5 − 2) × 180° = 540°, each 540° ÷ 5 = 108°; 72° is its angle at the centre and its exterior angle.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. C. No: the sides are equal, but not all the angles are
Neighbouring angles are 68° and 180° − 68° = 112°, not equal, so it isn't regular (a square would be).
14. 540, 109
Angle sum = (5 − 2) × 180° = 540° (∠ sum of polygon); the fifth = 540° − 121° − 99° − 118° − 93° = 109°.
15. 72, 360
At each corner it turns through an exterior angle, 360° ÷ 5 = 72°; one lap is one full turn, 360° (sum of ext. ∠s of polygon).
16. 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.
17. 21, 120
△OAB has three sides of 7 cm, perimeter 3 × 7 = 21 cm; ∠ABC is two equilateral triangles' 60°: 120°.
18. 72°
ED = EA and ∠E = 108°, so ∠EDA = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △); ∠ADC = 108° − 36° = 72°.
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