1. Mr Barton’s Maths Notes
Shape and Space
2. Polygons

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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
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

8. 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 =
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 = ÷ 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 = – Size of each exterior angle

9. 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

10. Good luck with your revision!