1. GCSE :: Interior & Exterior Angles
Dr J Frost
Objectives: Determine interior/exterior angles of a polygon, the number of sides of a polygon and consider tessellation of shapes.
Last modified: 24th December 2018

2. www.drfrostmaths.com Everything is completely free. Why not register?
Register now to interactively practise questions on this topic, including past paper questions and extension questions (including UKMT).
Teachers: you can create student accounts (or students can register themselves), to set work, monitor progress and even create worksheets.
With questions by:
Dashboard with points, trophies, notifications and student progress.
Questions organised by topic, difficulty and past paper.
Teaching videos with topic tests to check understanding.

3. Overview
We will consider the angles inside and outside regular and irregular polygons.
Types of questions we will explore:
What is each angle inside a regular decagon (10 sides)?
If the each angle inside a regular polygon is 175°, how many sides does it have?
If I put a bunch of regular pentagons in a circle, what shape would be formed in the middle? ?
Possible lesson structure:
Lesson 1: Interior angles.
Lesson 2: Regular Polygons & Exterior angles.
Lesson 3: Tessellation and Problem Solving.

4. STARTER :: Interior angles of quadrilateral
An interior angle is just an angle between two sides within the shape.
The interior angles of a quadrilateral add up to 360. ?
Parallelogram 1 2 𝑦
100° 𝑥
50° 𝑥
𝑥=130° ?
𝑥=100° 𝑦=80° ? ? 3 4
Trapezium
Kite 𝑥
𝑥=120° ? 𝑥
95°
55°
60°
Cointerior angles (“C” angles) add to 180°.
𝑥=105° ?

5. Sum of interior angles 𝑛=𝟑 𝑛=𝟒 Total of interior angles = 𝟑𝟔𝟎°
Can you guess what the angles add up to in a pentagon? How would you prove it?

6. ! For an 𝑛-sided shape, the sum of the interior angles is:
Sum of interior angles
Click to Fromanimate
We can cut a pentagon into three triangles. Notice that the angles combined in the three triangles are the total interior angle in the pentagon.
The sum of the interior angles of the triangles is:
3×180°=540°
! For an 𝑛-sided shape, the sum of the interior angles is:
180(𝑛−2) ?
(Notice the number of triangles we can form is 2 less than the number of sides. For a pentagon it was 5−2=3 triangles)

7. Examples
What is the total interior, and therefore, each interior angle, of each of these regular polygons? 1 2
Fro Tip: Memorise these for pentagon, hexagon and octagon.
70°
Determine 𝑥. 𝑥
Regular Polygon
Total interior
Each Interior Angle
Equilateral triangle (3)
𝟏𝟖𝟎×𝟏=𝟏𝟖𝟎°
𝟏𝟖𝟎°÷𝟑=𝟔𝟎°
Square (4)
𝟏𝟖𝟎×𝟐=𝟑𝟔𝟎°
𝟑𝟔𝟎°÷𝟒=𝟗𝟎°
Pentagon (5)
𝟏𝟖𝟎×𝟑=𝟓𝟒𝟎°
𝟓𝟒𝟎°÷𝟓=𝟏𝟎𝟖°
Hexagon (6)
𝟏𝟖𝟎×𝟒=𝟕𝟐𝟎°
𝟕𝟐𝟎°÷𝟔=𝟏𝟐𝟎°
Octagon (8)
𝟏𝟖𝟎×𝟔=𝟏𝟎𝟖𝟎°
𝟏𝟎𝟖𝟎°÷𝟖=𝟏𝟑𝟓°
Nonagon (9)
𝟏𝟖𝟎×𝟕=𝟏𝟐𝟔𝟎°
𝟏𝟐𝟔𝟎°÷𝟗=𝟏𝟒𝟎°
Decagon (10)
𝟏𝟖𝟎×𝟖=𝟏𝟒𝟒𝟎°
𝟏𝟒𝟒𝟎°÷𝟏𝟎=𝟏𝟒𝟒°
60°
110°
130° ?
6-sided so total interior angle:
𝟏𝟖𝟎× 𝟔−𝟐 =𝟕𝟐𝟎°
𝒙=𝟕𝟐𝟎− 𝟗𝟎+𝟏𝟏𝟎+𝟏𝟑𝟎+𝟔𝟎+𝟕𝟎 =𝟕𝟐𝟎°−𝟒𝟔𝟎° =𝟐𝟔𝟎° ? ? ? ? ? ? ? ? ? ?

8. Test Your Understanding
A regular dodecagon (12 sides).
160° 𝑥 𝑥
130°
80°
120°
𝑥=140° ?
𝑥=144° ?
120° 𝑥
40°
100°
𝑥=240° ?
40°

9. Exercise 1 ? ? ? ? ? ? (On supplied sheet) 1a b c b = 260 x = 75f
a = 100 ?
x = 222 ?
x = 309 ?

10. Exercise 1 ? ? ? ? ? g h i x = 120 x = 252 x = 54
The total of the interior angles of a polygon is 1260°. How many sides does it have?
The interior angle of a regular polygon is 179°. How many sides does it have? 2 ? N1 ?

11. Exercise 1
[UKMT] If a 𝑛-sided polygon has exactly 3 obtuse angles (i.e. 90°<𝜃<180°), the rest acute or right-angled, then determine the possible values of 𝑛 (Hint: determine the possible range for the sum of the interior angles, and use these inequalities to solve). N2 ?

12. Exterior Angles NO YES NO ? ? ? ?
An exterior angle of a polygon is an angle between the line extended from one side, and an adjacent side.
! At any vertex (i.e. corner):
𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟+𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟=180°
exterior
interior ?
Which of these are exterior angles of the polygon? ? ? NO
YES ? NO

13. Click to Start Damonimation
Total Exterior Angle
To defeat Kim Jon Il, Matt Damon must encircle his pentagonal palace.
What angle does Matt Damon turn in total?
He does one full rotation, so…
𝟑𝟔𝟎° ?
! The sum of the exterior angles of any polygon is 360°.
Click to Start Damonimation

14. Each Exterior/Interior Angle of Regular Polygon
If the pentagon is regular, then each exterior angles is the same. Therefore:
Each exterior angle of pentagon
𝟑𝟔𝟎 𝟓 =𝟕𝟐°
Each interior angle of pentagon
=𝟏𝟖𝟎−𝟕𝟐=𝟏𝟎𝟖° ?
𝟏𝟎𝟖°
𝟕𝟐° ?
This therefore gives us an alternative way to work out the interior angle of a regular polygon:
Previous method:
Total interior
=180 5−2 =540°
Each interior:
540÷5=108°
Alternative method:
Each exterior:
=360÷5=72°
Each interior:
=180−72=108°

15. Practicing Angles in Regular Polygons
Sides
Exterior Angle
Interior Angle 3
Equilateral Triangle
360 3 =120°
180−120=60° 4
Square
360 4 =90°
180−90=90° 5
Pentagon
360 5 =72°
180−72=108° 6
Hexagon
360 6 =60°
180−60=120° 8
Octagon
360 8 =45°
180−45=135° 9
Nonagon
360 9 =40°
180−40=140° 10
Decagon
=36°
180−36=144° ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Bonus Question: What is the largest number of sides a shape can have such that its interior angle is an integer?
360 sides. The interior angle will be 179°. ?

16. Number of sides of a regular polygon
The advantage of this new method is that it works backwards: we can easily work out the number of sides of the polygon given an exterior or interior angle.
Key formulae:
Total exterior angle =360°
Interior + Exterior =180°
The exterior angle of a regular polygon is 15°. How many sides does it have? ?
𝟑𝟔𝟎÷𝒏=𝟏𝟓°
Therefore 𝒏=𝟑𝟔𝟎÷𝟏𝟓=𝟐𝟒 sides
interior
exterior
The interior angle of a regular polygon is 144°. How many sides does it have? ?
Exterior angle =𝟏𝟖𝟎−𝟏𝟒𝟒=𝟑𝟔°
𝒏=𝟑𝟔𝟎°÷𝟑𝟔°=𝟏𝟎 sides

17. Quickfire Questions ? ? ? ? ? Exterior angle =18° 𝒏= 𝟑𝟔𝟎 𝟏𝟖 =𝟐𝟎
Your teacher will fire these questions at a variety of you – try to do in your head if you can!
Key formulae:
Total exterior angle =360°
Interior + Exterior =180°
Exterior angle =18°
𝒏= 𝟑𝟔𝟎 𝟏𝟖 =𝟐𝟎 ?
interior
exterior
Interior angle =160°
Ext =𝟏𝟖𝟎−𝟏𝟔𝟎=𝟐𝟎° 𝒏= 𝟑𝟔𝟎 𝟐𝟎 =𝟏𝟖 ?
Exterior angle =45°
𝒏= 𝟑𝟔𝟎 𝟒𝟓 =𝟖 ?
Ext =𝟏𝟖𝟎−𝟏𝟕𝟓=𝟓° 𝒏= 𝟑𝟔𝟎 𝟓 =𝟕𝟐
Interior angle =175° ?
Ext =𝟏𝟖𝟎−𝟏𝟓𝟎=𝟑𝟎° 𝒏= 𝟑𝟔𝟎 𝟑𝟎 =𝟏𝟐
Interior angle =150° ?

18. Test Your Understanding
Determine the exterior angle of a 40-sided regular polygon.
𝟑𝟔𝟎 𝟒𝟎 =𝟗° 1 4
50°
85°
75° ?
𝑐=70° ?
80° 𝑐
Determine the interior angle of a 15-sided regular polygon.
𝑬𝒙𝒕= 𝟑𝟔𝟎 𝟏𝟓 =𝟐𝟒° 𝑰𝒏𝒕=𝟏𝟖𝟎−𝟐𝟒=𝟏𝟓𝟔° 2 5 ? 𝑥 3
The interior angle of a regular polygon is 178°. Determine how many sides it has.
𝑬𝒙𝒕=𝟏𝟖𝟎−𝟏𝟕𝟖=𝟐° 𝒏= 𝟑𝟔𝟎 𝟐 =𝟏𝟖𝟎 sides
[Edexcel] The diagram shows a regular hexagon and a regular octagon.
Calculate the size of the angle marked 𝑥.
You must show all your working.
Angles around a point sum to 𝟑𝟔𝟎°.
𝒙=𝟑𝟔𝟎−𝟏𝟐𝟎−𝟏𝟑𝟓=𝟏𝟎𝟓° ? ?

19. Exercise 2 4
Determine how many sides a regular polygon with the following exterior angle would have:
30 sides
45 8 sides
12 30 sides
9 40 sides
[Edexcel IGCSE Jan2014(R)-3H Q3b] The diagram shows a regular 6-sided polygon. Work out the value of 𝑦.
𝟑𝟔𝟎 𝟔 =𝟔𝟎° 1 ? ? ? ? ?
Determine how many sides a regular polygon with the following interior angle would have:
156 15 sides
162 20 sides
144 10 sides
175 72 sides
[Edexcel IGCSE Jan2014(R)-3H Q3a] The diagram shows a regular 5-sided polygon. Work out the value of 𝑥.
𝟏𝟖𝟎− 𝟑𝟔𝟎 𝟓 =𝟏𝟎𝟖° 5 2 ? ? ? ? ? 3 6
𝒂=𝟑𝟔𝟎−𝟗𝟎−𝟖𝟎−𝟖𝟎=𝟏𝟎𝟎° ?
[Edexcel GCSE Mar13-1H Q13] The diagram shows a square and 4 regular pentagons. Work out the size of the angle marked 𝑥.
𝟑𝟔𝟎−𝟗𝟎−𝟏𝟎𝟖−𝟏𝟎𝟖 =𝟓𝟒° 𝑥 𝑎
80°
80° ?

20. Exercise 2
[JMO 2007 A6]
The sizes in degrees of the interior angles of a pentagon are consecutive whole numbers.
What is the size of the largest of these angles?
Total interior angle of pentagon:
𝟏𝟖𝟎 𝟓−𝟐 =𝟓𝟒𝟎°
𝟓𝟒𝟎 𝟓 =𝟏𝟎𝟖°
If we space out angles evenly around 𝟏𝟎𝟖°, they will still add to 𝟓𝟒𝟎°:
𝟏𝟎𝟔,𝟏𝟎𝟕,𝟏𝟎𝟖,𝟏𝟎𝟗,𝟏𝟏𝟎
So largest angle is 𝟏𝟏𝟎°. 9 7
[Edexcel IGCSE May2015(R)-3H Q2b]
The diagram shows 3 identical regular pentagons. Work out the value of 𝑦.
𝟑𝟔𝟎− 𝟑×𝟏𝟎𝟖 =𝟑𝟔° ? ? 8
[Edexcel June2007-5H Q15] 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon and 𝐴𝐵𝑄𝑃 is a square. Angle 𝐶𝐵𝑄=𝑥°. Work out the value of 𝑥°.
𝒙=𝟑𝟔𝟎−𝟏𝟐𝟎−𝟗𝟎=𝟏𝟓𝟎°
[IMC 2015 Q9] What is the value of 𝑝+𝑞+𝑟+𝑠+𝑡+𝑢+𝑣+𝑤+𝑥+𝑦 in the diagram?
We have two lots of exterior angles:
𝟑𝟔𝟎×𝟐=𝟕𝟐𝟎° 10 ? ?

21. Problem Solving with Interior/Exterior Angles
There are variety of skills that harder questions involving interior/exterior angles might involve:
#1: Tessellation
#2: Using isosceles triangles
Shapes ‘tessellate’ if they fit together, without overlap, to form a repeating pattern. B 𝐴 𝐵 C C A A 𝐷 𝐶 B B B B A A C 𝐸 𝐻 B B 𝐹 𝐺
“The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has.”
“𝐴𝐵𝐶𝐷 is a square and 𝐶𝐷𝐸𝐹𝐺𝐻 is a regular hexagon. Determine the angle 𝐶𝐵𝐻.”

22. Problem Solving with Interior/Exterior Angles
There are variety of skills that harder questions involving interior/exterior angles might involve:
#3: Dealing with a mixture of both irregular and regular polygons.
[IMC 2006 Q19] The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value of 𝑥?

23. #1 :: Tessellation
Key point: If shapes fit together, then the interior angles around any point where the shapes meet add to 360°.
[Edexcel GCSE Nov2012-1H Q18] The pattern is made from two types of tiles, tile A and tile B.
Both tile A and tile B are regular polygons.
Work out the number of sides tile A has.
This angle:
𝟔𝟎° (interior angle of equilateral triangle)
Therefore interior angle of tile A:
𝟑𝟔𝟎−𝟔𝟎 𝟐 =𝟏𝟓𝟎°
Exterior angle =𝟏𝟖𝟎−𝟏𝟓𝟎=𝟑𝟎°
𝒏= 𝟑𝟔𝟎 𝟑𝟎 =𝟏𝟐 sides ? ?
We’ve recently seen a method for finding the number of sides of a regular polygon given its interior angle! ? ?

24. Further Example ? B A C Interior angle of 𝑨: 𝟏𝟖𝟎− 𝟑𝟔𝟎 𝟔 =𝟏𝟐𝟎°
The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has.
𝟏𝟐𝟎°
𝟗𝟎°
𝟏𝟓𝟎° ?
Interior angle of 𝑨: 𝟏𝟖𝟎− 𝟑𝟔𝟎 𝟔 =𝟏𝟐𝟎°
Therefore interior angle of 𝑪=𝟑𝟔𝟎−𝟗𝟎−𝟏𝟐𝟎=𝟏𝟓𝟎°
Exterior angle of 𝑪=𝟏𝟖𝟎−𝟏𝟓𝟎=𝟑𝟎°
𝒏= 𝟑𝟔𝟎 𝟑𝟎 =𝟏𝟐

25. Test Your Understanding?
The diagram shows 4 congruent regular pentagons that form the sides of an 𝑛-sided regular polygon. Determine the value of 𝑛.
𝟏𝟎𝟖°
𝟏𝟎𝟖°
𝟏𝟒𝟒° ?
Interior angle of pentagon: 𝟏𝟖𝟎− 𝟑𝟔𝟎 𝟓 =𝟏𝟎𝟖°
Interior angle of large polygon:
𝟑𝟔𝟎−𝟏𝟎𝟖−𝟏𝟎𝟖=𝟏𝟒𝟒°
Exterior angle of large polygon:
𝟏𝟖𝟎−𝟏𝟒𝟒=𝟑𝟔°
𝒏= 𝟑𝟔𝟎 𝟑𝟔

26. #2 :: Irregular Polygons within Regular Polygons
[Edexcel GCSE June2016-2H Q12] The diagram shows a regular pentagon. 𝐴𝐵 and 𝐶𝐷 are two of the lines of symmetry of the pentagon. Work out the size of the angle marked 𝑥.
90° ? ?
90°
108° ? ?
108°
Total interior angle of this irregular polygon:
𝟏𝟖𝟎 𝟓−𝟐 =𝟓𝟒𝟎°
𝒙=𝟓𝟒𝟎− 𝟗𝟎+𝟗𝟎+𝟏𝟎𝟖+𝟏𝟎𝟖 =𝟏𝟒𝟒° ?
One method: Within the regular polygon is an irregular polygon for which we can determine each of the angles.
Recall that total interior of 𝑛-sided polygon: 180 𝑛−2 ?

27. Test Your Understanding
[IMC 2006 Q19] The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value of 𝑥?
The central shape is a pentagon.
Its total interior angle =𝟓𝟒𝟎°
𝒙=𝟓𝟒𝟎−𝟏𝟎𝟖−𝟏𝟎𝟖−𝟏𝟐𝟎−𝟏𝟐𝟎 =𝟖𝟒° ?
108° ?
120° ? ?
108° ?
120° ?

28. #3 :: Isosceles Triangles
When you have regular polygons in contact with each other, then their lengths are all the same. This regularly results in isosceles triangles. 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝐺 𝐻 ?
Since 𝑪𝑫𝑬𝑭𝑮𝑯 is regular, 𝑪𝑫=𝑪𝑯 and since 𝑨𝑩𝑪𝑫 is regular, 𝑪𝑫=𝑩𝑪. Therefore 𝑩𝑪=𝑪𝑯 and so triangle 𝑩𝑪𝑯 is isosceles.
∠𝑩𝑪𝑯=𝟑𝟔𝟎−𝟗𝟎−𝟏𝟐𝟎=𝟏𝟓𝟎° (angles around point 𝑪 sum to 𝟑𝟔𝟎°)
Therefore ∠𝑪𝑩𝑯= 𝟏𝟖𝟎−𝟏𝟓𝟎 𝟐 =𝟏𝟓°
“𝐴𝐵𝐶𝐷 is a square and 𝐶𝐷𝐸𝐹𝐺𝐻 is a regular hexagon. Determine the angle 𝐶𝐵𝐻.”

29. Test Your Understanding
[IMC 2009 Q12] The diagram shows a square inside a regular hexagon. What is the size of the marked angle at 𝑋? ?
𝑨𝑩𝑿 is isosceles (as with previous question)
∠𝑪𝑩𝑿=𝟏𝟎𝟖° ∴∠𝑨𝑩𝑿=𝟏𝟎𝟖−𝟗𝟎=𝟏𝟖°
As 𝑨𝑩𝑿 is isosceles:
∠𝑨𝑿𝑩= 𝟏𝟖𝟎−𝟏𝟖 𝟐 =𝟖𝟏° 𝐴
81° 𝑋
81°
108°
18°
90° 𝐶 𝐵

30. Exercise 3 ? ? ? ? 3 1 4 2 [Edexcel IGCSE Nov-2010-4H Q13]
The size of each interior angle of a regular polygon is 11 times the size of each exterior angle. Work out the number of sides the polygon has.
Interior : Exterior = 11 : 1. Exterior = 𝟏𝟖𝟎°÷𝟏𝟐°=𝟏𝟓 → 𝒏= 𝟑𝟔𝟎 𝟏𝟓 =𝟐𝟒
[KS3 SATs 2004 L6-L8 Paper 2 Q19 Edited]
A pupil has three tiles. One is a regular octagon, one is a regular hexagon, and one is a square. The side length of each tile is the same. The pupil says the hexagon will fit exactly like this. Is the pupil correct?
No, as 𝟏𝟑𝟓+𝟗𝟎+𝟏𝟐𝟎=𝟑𝟒𝟓° which is not 𝟑𝟔𝟎°. 1 ? 4 ? 2
[Edexcel GCSE Nov2014-1H Q17] ABCDEFGH is a regular octagon. BCKFGJ is a hexagon. JK is a line of symmetry of the hexagon. Angle 𝐵𝐽𝐺= angle 𝐶𝐾𝐹=140°. Work out the size of angle KFE.
Total interior angle of 𝑩𝑪𝑲𝑭𝑮𝑱=𝟏𝟖𝟎 𝟔−𝟐 =𝟕𝟐𝟎°.
∠𝑲𝑭𝑮= 𝟕𝟐𝟎−𝟏𝟒𝟎−𝟏𝟒𝟎 𝟒 =𝟏𝟏𝟎° → ∠𝑲𝑭𝑬=𝟏𝟑𝟓−𝟏𝟏𝟎=𝟐𝟓°
[Edexcel IGCSE Nov2009-3H Q3a]
The diagram shows a regular octagon, with centre O. Work out the value of 𝑥.
Half an interior angle of octagon: 𝟏𝟑𝟓 𝟐 =𝟔𝟕.𝟓° ? ?

31. Exercise 3 5
A regular polygon 𝐴 is surrounded by squares and equilateral triangles in an alternating pattern, as shown. Show that 𝐴 is a hexagon.
Interior angle of 𝑨=𝟑𝟔𝟎−𝟗𝟎−𝟗𝟎−𝟔𝟎=𝟏𝟐𝟎°
Exterior angle =𝟏𝟖𝟎−𝟏𝟐𝟎=𝟔𝟎°
𝒏= 𝟑𝟔𝟎 𝟔𝟎 =𝟔 so a hexagon. ? 𝐴
A regular polygon 𝐵 with 𝑛 sides is surrounded by squares and regular pentagons in an alternating pattern, as shown. Determine the value of 𝑛.
Interior angle of 𝑩=𝟑𝟔𝟎−𝟗𝟎−𝟏𝟎𝟖=𝟏𝟔𝟐°
Exterior angle =𝟏𝟖𝟎−𝟏𝟔𝟐=𝟏𝟖°
𝒏= 𝟑𝟔𝟎 𝟏𝟖 =𝟐𝟎 sides 6 ? 𝐵

32. Exercise 3 9
[IMC 2005 Q14] Ten stones, of identical shape and size, are used to make an arch, as shown in the diagram. Each stone has a cross-section in the shape of a trapezium with three equal sides. What is the size of the smallest angles of the trapezium? 7
[IMC 2003 Q22] The diagram shows a regular dodecagon (a polygon with twelve equal sides and equal angles). What is the size of the marked angle?
𝟕𝟓° ?
[JMO 2014 B1]
The figure shows an equilateral triangle ABC, a square BCDE, and a regular pentagon BEFGH. What is the difference between the sizes of ∠ADE and ∠AHE? 8 ?
𝟗𝟗° 10
[IMC 2018 Q18] The diagram shows a regular pentagon and an equilateral triangle placed inside a square. What is the value of 𝑥?
𝟐𝟒° (e.g. by first considering the angles in the bottom irregular pentagon) ? 𝟎° ?

33. Exercise 2 11
Find all regular polygons which tessellate (when restricted only to one type of polygon).
Equilateral triangle, square, hexagon.
By thinking about interior angles, prove that the regular polygons you identified above are the only regular polygons which tessellate.
Method 1: The possible exterior angles of a regular polygons are the factors of 360 less than 180: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120
This gives interior angles of 179, 178, ..., 140, 135, 120, 108, 90, 60.
To tessellate, the interior angle has to divide 360. Only 120, 90 and 60 does. This corresponds to a hexagon, square and equilateral triangle.
Method 2: 360 divided by the interior angle must give a whole number, in order for the regular polygon to tessellate. Interior angle is 𝟏𝟖𝟎− 𝟑𝟔𝟎 𝒏 , so 𝟑𝟔𝟎 𝟏𝟖𝟎− 𝟑𝟔𝟎 𝒏 =𝒌 for some constant 𝒌. Simplifying this gives 𝒌𝒏−𝟐𝒌−𝟐𝒏=𝟎
This factorises to 𝒌−𝟐 𝒏−𝟐 =𝟒
This only numbers which multiply to give 4 are 𝟏×𝟒 or 𝟐×𝟐 or 𝟒×𝟏. This 𝒏=𝟔,𝟒 or 𝟑 in each case. ? N ?

34. FINAL TOPIC REVIEW
Vote with your diaries! A B C D

35. What is the total exterior angle of a polygon in terms of the number of sides n?
360
360 n
360n
180(n-2)

36. What is the total interior angle of a 20 sided polygon?
360
3600
3240
6480

37. The interior angle of a polygon is 178. How many sides does it have?20 40 90
180

38. What is the interior angle of a 90 sided regular polygon?
172
176
178
179

39. Determine the angle .
61
29
105
120
215
223
225
235