Mr Barton’s Maths Notes Shape and Space 2. Polygons www.mrbartonmaths.com

2. Polygons One of Mr Barton’s Top 10 Maths Jokes What are Polygons? What did the pirate (who was also a very keen mathematician) say when his parrot flew away?... “Poly-gon!”… you can’t beat a maths joke, hey?... anyway… What are Polygons? A Polygon is any closed shape which has three or more sides. Regular Polygons All their sides are the same length, and all their angles are the same size e.g. squares, equilateral triangles, regular octagons… Irregular Polygons You’ve guessed it… these do not have equal length sides and angles Rectangle, kites and trapeziums are an irregular polygons, but so too are shapes like this: Two types of Polygons that you must be especially clued up about are quadrilaterals and triangles

1. Triangles There are 4 types of triangles you need to be on the look-out for and you must know the properties of (what is special about) each of them All angles are equal (600 each) All sides are the same length Three lines of symmetry Equilateral Two angles are equal Two sides are the same length One line of symmetry Isosceles One angle is 900 All sides may be different lengths All angles may be different May have 0 or 1 line of symmetry Right Angled All angles are different sizes All sides are different lengths No lines of symmetry Scalene

2. Quadrilaterals A Quadrilateral is any four-sided shape. There are lots of quadrilaterals flying around, and it is important that you know the properties of each… so here they are! All angles are right-angles (900 each) All sides are the same length Two pairs of parallel lines Four lines of symmetry Square Opposite angles are equal Opposite sides are the same length Two pairs of parallel sides May have no lines of symmetry Parallelogram All angles are right-angles (900 each) Opposite sides are the same length Opposite sides are parallel Has two lines of symmetry Rectangle

Opposite angles are equal All sides are the same length Opposite sides are parallel Two lines of symmetry Rhombus Notice: Each of the four shapes above are very similar… in fact, they are all just special types of parallelograms! See how they each have two pairs of parallel sides… and then it just certain other properties that make them different shapes! All angles may be different sizes All sides may be different lengths Opposite sides are parallel May have no lines of symmetry Trapezium One pair of equal angles Adjacent sides are the same length No pairs of parallel sides One line of symmetry Kite

3. Other Polygons As soon as you get above 4 sides, the names of the polygons start to get a bit weird. Here are some of the main ones you should learn. Notice: Each of the shapes below are regular polygons as all the sides and angles are the same… but any 8 sided shape is still an octagon, it may just be an irregular one! 8 sides 5 sides 6 sides 7 sides Pentagon Hexagon Heptagon / Heptagon Octagon 20 sides 9 sides 10 sides 12 sides Nonagon Decagon Dodecagon Icosagon

4. Interior Angles of Polygons An interior angle is any angle inside the polygon If we are told the number of sides a polygon has, we can work out the total sum of all the interior angles using this little formula: Sum of all interior angles = (Number of sides of polygon – 2) x 180 Why? Well, it’s all to do with triangles… We know that the sum of the interior angles of any triangle is 1800, right? Well… we can split any polygon up into triangles, like this… And there will always be 2 fewer triangles than there are sides! 2 3 1 4 6 sides 4 triangles For Regular Polygons Because all angles are equal in regular polygons, you can work out the size of each interior angle like this: Size of each interior angle = Sum of all interior angles ÷ Number of sides

5. Exterior Angles of Polygons An exterior angle is an angle outside the polygon made by extending one of the sides… And here is the fact! exterior angle Sum of all exterior angles = 3600 Why? Well, if you keep moving around the polygon, extending the sides and measuring each exterior angle, by the time you get back to where you started you have made… a circle! Which, as we all know, contains 3600 For Regular Polygons If all interior angles are equal for regular polygons, then all exterior angles are equal too, so to work out the size of each one, we do this… Size of each exterior angle = 3600 ÷ Number of sides Note: If you know the sizes of the exterior angles of a regular polygon, then you can also work out the sizes of the interiors by remembering that angles on a straight line add up to 1800 Size of each interior angle = 1800 – Size of each exterior angle

6. Massive Table of Facts Using the formulae we have talked about, it is possible to work out pretty much any angle fact about any size polygon. Have a practice to make sure you can get the numbers in this table… Name of Polygon Number of Sides Total Sum of Interior Angles Size of each Interior Angle if Regular Total Sum of Exterior Angles Size of each Exterior Angle if Regular Triangle 3 180 60 360 120 Quadrilateral 4 90 Pentagon 5 540 108 72 Hexagon 6 720 Heptagon 7 900 128.6 (1dp) 51.4 (1dp) Octagon 8 1080 135 45

Good luck with your revision!