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Mathematics questions, chapter 4
Nama / NameKelas / ClassTarikh / Date

Polygon

44 marks. Use Practice to check each answer as you go, or Exam to answer everything first. The diagrams are drawn to scale, but you must calculate the angles, not measure them. Show your working for Section B.

Section A: objective questions

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Which of the following is a regular polygon?
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C
A regular polygon has all sides equal and all interior angles equal. An equilateral triangle has three equal sides and three 60° angles. A rectangle has equal angles only, and a rhombus has equal sides only.
For the teacher: Remembering, low difficulty, standard 4.1.1
Textbook p. 56
2

Question 2

How many axes of symmetry does a regular octagon have?
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B
A regular polygon has as many axes of symmetry as it has sides, so a regular octagon has 8. The number 16 comes from counting both ends of each axis.
For the teacher: Remembering, low difficulty, standard 4.1.1
Textbook p. 58
3

Question 3

A rhombus that is not a square has four equal sides. Why is it not a regular polygon?
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B
A regular polygon needs equal sides and equal interior angles. A rhombus has equal sides, but its angles are not all equal (two are acute and two are obtuse). It does have 4 sides and 2 axes of symmetry.
For the teacher: Understanding, medium difficulty, standard 4.1.1
Textbook p. 56
4

Question 4

Find the sum of the interior angles of a decagon (10 sides).
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D
(10 − 2) × 180° = 8 × 180° = 1 440°. 1 800° forgets to subtract 2, and 1 260° and 1 620° use 7 and 9 triangles.
For the teacher: Applying, low difficulty, standard 4.2.1
Textbook p. 63
5

Question 5

What is the size of one interior angle of a regular nonagon (9 sides)?
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A
Exterior angle = 360° ÷ 9 = 40°, so the interior angle = 180° − 40° = 140°. 40° is the exterior angle, and 135° is the interior angle of an octagon.
For the teacher: Applying, low difficulty, standard 4.2.3
Textbook p. 65
6

Question 6

What is the exterior angle of a regular polygon with 18 sides?
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C
Exterior angle = 360° ÷ 18 = 20°. 160° is the interior angle (180° − 20°).
For the teacher: Applying, low difficulty, standard 4.2.3
Textbook p. 65
7

Question 7

The exterior angle of a regular polygon is 15°. How many sides does the polygon have?
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D
n = 360° ÷ 15° = 24. Check the others: 12 sides gives 30°, 15 sides gives 24° and 20 sides gives 18°.
For the teacher: Applying, low difficulty, standard 4.2.3
Textbook p. 66
8

Question 8

Diagram 1 shows a quadrilateral. Find the value of x.
Quadrilateral with interior angles 85 degrees, 110 degrees, 70 degrees and x
Diagram 1
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A
The interior angles of a quadrilateral add up to (4 − 2) × 180° = 360°. x = 360° − 85° − 110° − 70° = 95°. 265° is the total of the three given angles.
For the teacher: Applying, low difficulty, standard 4.2.1
Textbook p. 64
9

Question 9

In a regular polygon, each interior angle is 5 times the exterior angle. How many sides does the polygon have?
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C
Interior + exterior = 180° and interior = 5 × exterior, so 6 × exterior = 180° and exterior = 30°. n = 360° ÷ 30° = 12.
For the teacher: Analysing, medium difficulty, standard 4.2.3
Textbook p. 66
10

Question 10

What is the sum of the exterior angles of an octagon?
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A
The sum of the exterior angles of any polygon is 360°. 1 080° is the sum of the interior angles, and 45° is one exterior angle of a regular octagon.
For the teacher: Understanding, low difficulty, standard 4.2.2
Textbook p. 64
11

Question 11

Diagram 2 shows a regular pentagon and a regular hexagon that share the side CD. Find the size of the angle at C between CB and CI (the angle that is outside both polygons).
A regular pentagon on the left and a regular hexagon on the right sharing a vertical side CD; B is the pentagon vertex next to C and I is the hexagon vertex next to C
Diagram 2
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D
The angles at C add up to 360°. Interior angle of the pentagon = 108°; interior angle of the hexagon = 120°. The outside angle = 360° − 108° − 120° = 132°. 228° is 108° + 120°.
For the teacher: Analysing, high difficulty, standard 4.2.4
Textbook p. 66
12

Question 12

Which of these angles cannot be an interior angle of a regular polygon?
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B
For 100° the exterior angle is 80° and n = 360 ÷ 80 = 4.5, which is not a whole number. 108° gives n = 5, 120° gives n = 6 and 140° gives n = 9.
For the teacher: Evaluating, medium difficulty, standard 4.2.3
Textbook p. 66

Section B: structured questions

32 marks
Answer all questions. Show your working.
13

Question 13

A regular polygon has 12 sides.
[1 mark] (a)State the name of the polygon.
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Tick each point you got.
[1 mark] (b)How many axes of symmetry does it have?
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Tick each point you got.
[1 mark] (c)Find the size of one exterior angle.
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Tick each point you got.
[1 mark] (d)Find the size of one interior angle.
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Tick each point you got.
For the teacher: Remembering, low difficulty, standard 4.2.3
Textbook p. 65
14

Question 14

Diagram 3 shows a pentagon.
Pentagon with interior angles 105, 130, 90, 115 degrees and x; the 90 degree angle has a right-angle mark
Diagram 3
[1 mark] (a)Find the sum of the interior angles of a pentagon.
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Tick each point you got.
[2 marks] (b)Find the value of x.
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Tick each point you got.
For the teacher: Applying, low difficulty, standard 4.2.1
Textbook p. 64
15

Question 15

Diagram 4 shows a hexagon with one side extended at every vertex. The exterior angles are 45°, 60°, 50°, 70°, x and 2x.
Hexagon with a side extended at each vertex; exterior angles 45, 60, 50, 70, x and 2x degrees
Diagram 4
[1 mark] (a)The sum of the exterior angles of this hexagon is
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360°
The exterior angles of any polygon add up to 360°. 720° is the sum of the interior angles of a hexagon.
[2 marks] (b)Find the value of x.
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Tick each point you got.
[1 mark] (c)Find the interior angle at the vertex whose exterior angle is 2x.
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Tick each point you got.
For the teacher: Applying, low difficulty, standard 4.2.2
Textbook p. 64
16

Question 16

In Diagram 5, ABCDEFGH is a regular octagon. AB is extended and DC is extended so that they meet at P.
Regular octagon ABCDEFGH with side AB extended beyond B and side DC extended beyond C, meeting at P below and to the right of BC
Diagram 5
[1 mark] (a)Find the exterior angle of the octagon.
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Tick each point you got.
[1 mark] (b)State ∠PBC and ∠PCB.
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Tick each point you got.
[1 mark] (c)Find ∠BPC.
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Tick each point you got.
[1 mark] (d)What type of triangle is PBC?
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PB = PC because the base angles are equal.
Tick each point you got.
For the teacher: Analysing, high difficulty, standard 4.2.4
Textbook p. 66
17

Question 17

Each interior angle of a regular polygon is 150°.
[1 mark] (a)Find the exterior angle.
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Tick each point you got.
[1 mark] (b)Find the number of sides of the polygon.
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Tick each point you got.
[2 marks] (c)Find the sum of the interior angles of the polygon.
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Check: 12 × 150° = 1 800°.
Tick each point you got.
For the teacher: Applying, low difficulty, standard 4.2.3
Textbook p. 66
18

Question 18

The sum of the interior angles of a regular polygon is 3 240°.
[2 marks] (a)Find the number of sides.
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Tick each point you got.
[1 mark] (b)State the name of this polygon.
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Tick each point you got.
[1 mark] (c)Find the size of one exterior angle.
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Tick each point you got.
For the teacher: Applying, low difficulty, standard 4.2.3
Textbook p. 66
19

Question 19

In a regular polygon, an interior angle is (5y + 10)° and an exterior angle is (y + 20)°.
[2 marks] (a)Find the value of y.
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Tick each point you got.
[1 mark] (b)Find the exterior angle and the interior angle.
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[2 marks] (c)Find the number of sides and name the polygon.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 4.2.3
Textbook p. 66
20

Question 20

Azreen designs a logo for her school club. It is a regular hexagon with the centre O joined to every vertex, as in Diagram 6. The distance from O to each vertex is 4 cm.
Regular hexagon with its centre O joined to all six vertices; one radius is labelled 4 cm
Diagram 6
[1 mark] (a)Find the angle at O between two neighbouring radii.
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Tick each point you got.
[2 marks] (b)Each triangle at the centre has two sides of 4 cm and an angle of 60° between them. What is the length of each side of the hexagon? Give a reason.
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Tick each point you got.
[1 mark] (c)Azreen draws the hexagon with compasses and a ruler. What should she set the compasses to, to mark the vertices round the circle?
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The side of a regular hexagon equals the radius of its circle.
Tick each point you got.
For the teacher: Creating, medium difficulty, standard 4.1.2
Textbook p. 60