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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
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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 175ยฐ, 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.
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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๏ฐ. ? Parallelogram 1 2 ๐ฆ 100ยฐ ๐ฅ 50ยฐ ๐ฅ ๐ฅ=130ยฐ ? ๐ฅ=100ยฐ ๐ฆ=80ยฐ ? ? 3 4 Trapezium Kite ๐ฅ ๐ฅ=120ยฐ ? ๐ฅ 95ยฐ 55ยฐ 60ยฐ Cointerior angles (โCโ angles) add to 180ยฐ. ๐ฅ=105ยฐ ?
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Sum of interior angles ๐=๐ ๐=๐ Total of interior angles = ๐๐๐ยฐ
Can you guess what the angles add up to in a pentagon? How would you prove it?
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! For an ๐-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: 3ร180ยฐ=540ยฐ ! For an ๐-sided shape, the sum of the interior angles is: 180(๐โ2) ? (Notice the number of triangles we can form is 2 less than the number of sides. For a pentagon it was 5โ2=3 triangles)
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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. 70ยฐ Determine ๐ฅ. ๐ฅ Regular Polygon Total interior Each Interior Angle Equilateral triangle (3) ๐๐๐ร๐=๐๐๐ยฐ ๐๐๐ยฐรท๐=๐๐ยฐ Square (4) ๐๐๐ร๐=๐๐๐ยฐ ๐๐๐ยฐรท๐=๐๐ยฐ Pentagon (5) ๐๐๐ร๐=๐๐๐ยฐ ๐๐๐ยฐรท๐=๐๐๐ยฐ Hexagon (6) ๐๐๐ร๐=๐๐๐ยฐ ๐๐๐ยฐรท๐=๐๐๐ยฐ Octagon (8) ๐๐๐ร๐=๐๐๐๐ยฐ ๐๐๐๐ยฐรท๐=๐๐๐ยฐ Nonagon (9) ๐๐๐ร๐=๐๐๐๐ยฐ ๐๐๐๐ยฐรท๐=๐๐๐ยฐ Decagon (10) ๐๐๐ร๐=๐๐๐๐ยฐ ๐๐๐๐ยฐรท๐๐=๐๐๐ยฐ 60ยฐ 110ยฐ 130ยฐ ? 6-sided so total interior angle: ๐๐๐ร ๐โ๐ =๐๐๐ยฐ ๐=๐๐๐โ ๐๐+๐๐๐+๐๐๐+๐๐+๐๐ =๐๐๐ยฐโ๐๐๐ยฐ =๐๐๐ยฐ ? ? ? ? ? ? ? ? ? ?
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Test Your Understanding
A regular dodecagon (12 sides). 160ยฐ ๐ฅ ๐ฅ 130ยฐ 80ยฐ 120ยฐ ๐ฅ=140ยฐ ? ๐ฅ=144ยฐ ? 120ยฐ ๐ฅ 40ยฐ 100ยฐ ๐ฅ=240ยฐ ? 40ยฐ
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Exercise 1 ? ? ? ? ? ? (On supplied sheet) 1a b c b = 260๏ฐ x = 75๏ฐ
f a = 100๏ฐ ? x = 222๏ฐ ? x = 309๏ฐ ?
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Exercise 1 ? ? ? ? ? g h i x = 120๏ฐ x = 252๏ฐ x = 54๏ฐ
The total of the interior angles of a polygon is 1260ยฐ. How many sides does it have? The interior angle of a regular polygon is 179ยฐ. How many sides does it have? 2 ? N1 ?
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Exercise 1 [UKMT] If a ๐-sided polygon has exactly 3 obtuse angles (i.e. 90ยฐ<๐<180ยฐ), the rest acute or right-angled, then determine the possible values of ๐ (Hint: determine the possible range for the sum of the interior angles, and use these inequalities to solve). N2 ?
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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): ๐๐๐ก๐๐๐๐๐+๐๐ฅ๐ก๐๐๐๐๐=180ยฐ exterior interior ? Which of these are exterior angles of the polygon? ? ? NO YES ? NO
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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, soโฆ ๐๐๐ยฐ ? ! The sum of the exterior angles of any polygon is 360ยฐ. Click to Start Damonimation
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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 ๐๐๐ ๐ =๐๐ยฐ Each interior angle of pentagon =๐๐๐โ๐๐=๐๐๐ยฐ ? ๐๐๐ยฐ ๐๐ยฐ ? This therefore gives us an alternative way to work out the interior angle of a regular polygon: Previous method: Total interior =180 5โ2 =540ยฐ Each interior: 540รท5=108ยฐ Alternative method: Each exterior: =360รท5=72ยฐ Each interior: =180โ72=108ยฐ
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Practicing Angles in Regular Polygons
Sides Exterior Angle Interior Angle 3 Equilateral Triangle 360 3 =120ยฐ 180โ120=60ยฐ 4 Square 360 4 =90ยฐ 180โ90=90ยฐ 5 Pentagon 360 5 =72ยฐ 180โ72=108ยฐ 6 Hexagon 360 6 =60ยฐ 180โ60=120ยฐ 8 Octagon 360 8 =45ยฐ 180โ45=135ยฐ 9 Nonagon 360 9 =40ยฐ 180โ40=140ยฐ 10 Decagon =36ยฐ 180โ36=144ยฐ ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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 179ยฐ. ?
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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 =360ยฐ Interior + Exterior =180ยฐ The exterior angle of a regular polygon is 15ยฐ. How many sides does it have? ? ๐๐๐รท๐=๐๐ยฐ Therefore ๐=๐๐๐รท๐๐=๐๐ sides interior exterior The interior angle of a regular polygon is 144ยฐ. How many sides does it have? ? Exterior angle =๐๐๐โ๐๐๐=๐๐ยฐ ๐=๐๐๐ยฐรท๐๐ยฐ=๐๐ sides
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Quickfire Questions ? ? ? ? ? Exterior angle =18ยฐ ๐= ๐๐๐ ๐๐ =๐๐
Your teacher will fire these questions at a variety of you โ try to do in your head if you can! Key formulae: Total exterior angle =360ยฐ Interior + Exterior =180ยฐ Exterior angle =18ยฐ ๐= ๐๐๐ ๐๐ =๐๐ ? interior exterior Interior angle =160ยฐ Ext =๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐๐ ? Exterior angle =45ยฐ ๐= ๐๐๐ ๐๐ =๐ ? Ext =๐๐๐โ๐๐๐=๐ยฐ ๐= ๐๐๐ ๐ =๐๐ Interior angle =175ยฐ ? Ext =๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐๐ Interior angle =150ยฐ ?
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Test Your Understanding
Determine the exterior angle of a 40-sided regular polygon. ๐๐๐ ๐๐ =๐ยฐ 1 4 50ยฐ 85ยฐ 75ยฐ ? ๐=70ยฐ ? 80ยฐ ๐ Determine the interior angle of a 15-sided regular polygon. ๐ฌ๐๐= ๐๐๐ ๐๐ =๐๐ยฐ ๐ฐ๐๐=๐๐๐โ๐๐=๐๐๐ยฐ 2 5 ? ๐ฅ 3 The interior angle of a regular polygon is 178ยฐ. Determine how many sides it has. ๐ฌ๐๐=๐๐๐โ๐๐๐=๐ยฐ ๐= ๐๐๐ ๐ =๐๐๐ sides [Edexcel] The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked ๐ฅ. You must show all your working. Angles around a point sum to ๐๐๐ยฐ. ๐=๐๐๐โ๐๐๐โ๐๐๐=๐๐๐ยฐ ? ?
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Exercise 2 4 Determine how many sides a regular polygon with the following exterior angle would have: 30๏ฐ sides 45๏ฐ 8 sides 12๏ฐ 30 sides 9๏ฐ 40 sides [Edexcel IGCSE Jan2014(R)-3H Q3b] The diagram shows a regular 6-sided polygon. Work out the value ofย ๐ฆ. ๐๐๐ ๐ =๐๐ยฐ 1 ? ? ? ? ? Determine how many sides a regular polygon with the following interior angle would have: 156๏ฐ 15 sides 162๏ฐ 20 sides 144๏ฐ 10 sides 175๏ฐ 72 sides [Edexcel IGCSE Jan2014(R)-3H Q3a] The diagram shows a regular 5-sided polygon. Work out the value of ๐ฅ. ๐๐๐โ ๐๐๐ ๐ =๐๐๐ยฐ 5 2 ? ? ? ? ? 3 6 ๐=๐๐๐โ๐๐โ๐๐โ๐๐=๐๐๐ยฐ ? [Edexcel GCSE Mar13-1H Q13] The diagram shows a square and 4 regular pentagons. Work out the size of the angle marked ๐ฅ. ๐๐๐โ๐๐โ๐๐๐โ๐๐๐ =๐๐ยฐ ๐ฅ ๐ 80ยฐ 80ยฐ ?
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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: ๐๐๐ ๐โ๐ =๐๐๐ยฐ ๐๐๐ ๐ =๐๐๐ยฐ If we space out angles evenly around ๐๐๐ยฐ, they will still add to ๐๐๐ยฐ: ๐๐๐,๐๐๐,๐๐๐,๐๐๐,๐๐๐ So largest angle is ๐๐๐ยฐ. 9 7 [Edexcel IGCSE May2015(R)-3H Q2b] The diagram shows 3 identical regular pentagons. Work out the value of ๐ฆ. ๐๐๐โ ๐ร๐๐๐ =๐๐ยฐ ? ? 8 [Edexcel June2007-5Hย Q15] ๐ด๐ต๐ถ๐ท๐ธ๐น is a regular hexagon and ๐ด๐ต๐๐ is a square. Angle ๐ถ๐ต๐=๐ฅยฐ. Work out the value ofย ๐ฅยฐ. ๐=๐๐๐โ๐๐๐โ๐๐=๐๐๐ยฐ [IMC 2015 Q9]ย What is the value ofย ๐+๐+๐+๐ +๐ก+๐ข+๐ฃ+๐ค+๐ฅ+๐ฆ in the diagram? We have two lots of exterior angles: ๐๐๐ร๐=๐๐๐ยฐ 10 ? ?
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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 โtessellateโ if they fit together, without overlap, to form a repeating pattern. B ๐ด ๐ต C C A A ๐ท ๐ถ B B B B A A C ๐ธ ๐ป B B ๐น ๐บ โThe above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has.โ โ๐ด๐ต๐ถ๐ท is a square and ๐ถ๐ท๐ธ๐น๐บ๐ป is a regular hexagon. Determine the angle ๐ถ๐ต๐ป.โ
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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 ofย ๐ฅ?
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#1 :: Tessellation Key point: If shapes fit together, then the interior angles around any point where the shapes meet add to 360ยฐ. [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: ๐๐ยฐ (interior angle of equilateral triangle) Therefore interior angle of tile A: ๐๐๐โ๐๐ ๐ =๐๐๐ยฐ Exterior angle =๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐๐ sides ? ? Weโve recently seen a method for finding the number of sides of a regular polygon given its interior angle! ? ?
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Further Example ? B A C Interior angle of ๐จ: ๐๐๐โ ๐๐๐ ๐ =๐๐๐ยฐ
The above repeating pattern consists of three regular polygons, A (hexagon), B (square) and C. Determine how many sides C has. ๐๐๐ยฐ ๐๐ยฐ ๐๐๐ยฐ ? Interior angle of ๐จ: ๐๐๐โ ๐๐๐ ๐ =๐๐๐ยฐ Therefore interior angle of ๐ช=๐๐๐โ๐๐โ๐๐๐=๐๐๐ยฐ Exterior angle of ๐ช=๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐๐
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Test Your Understanding
? The diagram shows 4 congruent regular pentagons that form the sides of an ๐-sided regular polygon. Determine the value of ๐. ๐๐๐ยฐ ๐๐๐ยฐ ๐๐๐ยฐ ? Interior angle of pentagon: ๐๐๐โ ๐๐๐ ๐ =๐๐๐ยฐ Interior angle of large polygon: ๐๐๐โ๐๐๐โ๐๐๐=๐๐๐ยฐ Exterior angle of large polygon: ๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐
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#2 :: Irregular Polygons within Regular Polygons
[Edexcel GCSE June2016-2H Q12] The diagram shows a regular pentagon. ๐ด๐ต and ๐ถ๐ท are two of the lines of symmetry of the pentagon. Work out the size of the angle marked ๐ฅ. 90ยฐ ? ? 90ยฐ 108ยฐ ? ? 108ยฐ Total interior angle of this irregular polygon: ๐๐๐ ๐โ๐ =๐๐๐ยฐ ๐=๐๐๐โ ๐๐+๐๐+๐๐๐+๐๐๐ =๐๐๐ยฐ ? One method: Within the regular polygon is an irregular polygon for which we can determine each of the angles. Recall that total interior of ๐-sided polygon: 180 ๐โ2 ?
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Test Your Understanding
[IMC 2006 Q19] The diagram shows a regular pentagon and a regular hexagon which overlap. What is the value ofย ๐ฅ? The central shape is a pentagon. Its total interior angle =๐๐๐ยฐ ๐=๐๐๐โ๐๐๐โ๐๐๐โ๐๐๐โ๐๐๐ =๐๐ยฐ ? 108ยฐ ? 120ยฐ ? ? 108ยฐ ? 120ยฐ ?
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#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. ๐ด ๐ต ๐ถ ๐ท ๐ธ ๐น ๐บ ๐ป ? Since ๐ช๐ซ๐ฌ๐ญ๐ฎ๐ฏ is regular, ๐ช๐ซ=๐ช๐ฏ and since ๐จ๐ฉ๐ช๐ซ is regular, ๐ช๐ซ=๐ฉ๐ช. Therefore ๐ฉ๐ช=๐ช๐ฏ and so triangle ๐ฉ๐ช๐ฏ is isosceles. โ ๐ฉ๐ช๐ฏ=๐๐๐โ๐๐โ๐๐๐=๐๐๐ยฐ (angles around point ๐ช sum to ๐๐๐ยฐ) Therefore โ ๐ช๐ฉ๐ฏ= ๐๐๐โ๐๐๐ ๐ =๐๐ยฐ โ๐ด๐ต๐ถ๐ท is a square and ๐ถ๐ท๐ธ๐น๐บ๐ป is a regular hexagon. Determine the angle ๐ถ๐ต๐ป.โ
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Test Your Understanding
[IMC 2009 Q12] The diagram shows a square inside a regular hexagon. What is the size of the marked angle atย ๐? ? ๐จ๐ฉ๐ฟ is isosceles (as with previous question) โ ๐ช๐ฉ๐ฟ=๐๐๐ยฐ โดโ ๐จ๐ฉ๐ฟ=๐๐๐โ๐๐=๐๐ยฐ As ๐จ๐ฉ๐ฟ is isosceles: โ ๐จ๐ฟ๐ฉ= ๐๐๐โ๐๐ ๐ =๐๐ยฐ ๐ด 81ยฐ ๐ 81ยฐ 108ยฐ 18ยฐ 90ยฐ ๐ถ ๐ต
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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 = ๐๐๐ยฐรท๐๐ยฐ=๐๐ โ ๐= ๐๐๐ ๐๐ =๐๐ [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 ๐๐๐+๐๐+๐๐๐=๐๐๐ยฐ which is not ๐๐๐ยฐ. 1 ? 4 ? 2 [Edexcel GCSE Nov2014-1H Q17] ABCDEFGHย is a regular octagon. BCKFGJย is a hexagon. JK is a line of symmetry of the hexagon. Angleย ๐ต๐ฝ๐บ= angle ๐ถ๐พ๐น=140ยฐ. Work out the size of angleย KFE. Total interior angle of ๐ฉ๐ช๐ฒ๐ญ๐ฎ๐ฑ=๐๐๐ ๐โ๐ =๐๐๐ยฐ. โ ๐ฒ๐ญ๐ฎ= ๐๐๐โ๐๐๐โ๐๐๐ ๐ =๐๐๐ยฐ โ โ ๐ฒ๐ญ๐ฌ=๐๐๐โ๐๐๐=๐๐ยฐ [Edexcel IGCSE Nov2009-3H Q3a] The diagram shows a regular octagon, with centre O. Work out the value of ๐ฅ. Half an interior angle of octagon: ๐๐๐ ๐ =๐๐.๐ยฐ ? ?
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Exercise 3 5 A regular polygon ๐ด is surrounded by squares and equilateral triangles in an alternating pattern, as shown. Show that ๐ด is a hexagon. Interior angle of ๐จ=๐๐๐โ๐๐โ๐๐โ๐๐=๐๐๐ยฐ Exterior angle =๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐ so a hexagon. ? ๐ด A regular polygon ๐ต with ๐ sides is surrounded by squares and regular pentagons in an alternating pattern, as shown. Determine the value of ๐. Interior angle of ๐ฉ=๐๐๐โ๐๐โ๐๐๐=๐๐๐ยฐ Exterior angle =๐๐๐โ๐๐๐=๐๐ยฐ ๐= ๐๐๐ ๐๐ =๐๐ sides 6 ? ๐ต
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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? ๐๐ยฐ ? [JMO 2014 B1] The figure shows an equilateral triangleย ABC, a squareย BCDE, and a regular pentagonย BEFGH. What is the difference between the sizes ofย โ ADEย andย โ AHE? 8 ? ๐๐ยฐ 10 [IMC 2018 Q18]ย The diagram shows a regular pentagon and an equilateral triangle placed inside a square. What is the value ofย ๐ฅ? ๐๐ยฐ (e.g. by first considering the angles in the bottom irregular pentagon) ? ๐ยฐ ?
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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 ๐๐๐โ ๐๐๐ ๐ , so ๐๐๐ ๐๐๐โ ๐๐๐ ๐ =๐ for some constant ๐. Simplifying this gives ๐๐โ๐๐โ๐๐=๐ This factorises to ๐โ๐ ๐โ๐ =๐ This only numbers which multiply to give 4 are ๐ร๐ or ๐ร๐ or ๐ร๐. This ๐=๐,๐ or ๐ in each case. ? N ?
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FINAL TOPIC REVIEW Vote with your diaries! A B C D
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What is the total exterior angle of a polygon in terms of the number of sides n?
360 360 n 360n 180(n-2)
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What is the total interior angle of a 20 sided polygon?
360 3600 3240 6480
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The interior angle of a polygon is 178. How many sides does it have?
20 40 90 180
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What is the interior angle of a 90 sided regular polygon?
172๏ฐ 176๏ฐ 178๏ฐ 179๏ฐ
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Determine the angle . 61๏ฐ 29๏ฐ 105๏ฐ 120๏ฐ 215๏ฐ 223๏ฐ 225๏ฐ 235๏ฐ
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