Maths Notes
Shape and Space
2. Polygons
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2. Polygons
One of Mr Barton’s Top 10 Maths Jokes
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 (60 each)
• All sides are the same length
• Three lines of symmetry
0
Equilateral
Isosceles
• Two angles are equal
• Two sides are the same length
• One line of symmetry
Right Angled
• One angle is 90
• All sides may be different lengths
• All angles may be different
• May have 0 or 1 line of symmetry
Scalene
• All angles are different sizes
• All sides are different lengths
• No lines of symmetry
0
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 (90
• All sides are the same length
• Two pairs of parallel lines
• Four lines of symmetry
0
Square
Parallelogram
• Opposite angles are equal
• Opposite sides are the same length
• Two pairs of parallel sides
• May have no lines of symmetry
• All angles are right-angles (90 each)
• Opposite sides are the same length
• Opposite sides are parallel
• Has two lines of symmetry
0
Rectangle
each)
Rhombus
• Opposite angles are equal
• All sides are the same length
• Opposite sides are parallel
• Two lines of symmetry
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!
Trapezium
Kite
• All angles may be different sizes
• All sides may be different lengths
• Opposite sides are parallel
• May have no lines of symmetry
• One pair of equal angles
• Adjacent sides are the same length
• No pairs of parallel sides
• One line of symmetry
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!
5 sides
6 sides
Pentagon
Hexagon
7 sides
Heptagon / Heptagon
9 sides
10 sides
12 sides
Nonagon
Decagon
Dodecagon
8 sides
Octagon
20 sides
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…
exterior
angle
And here is the fact!
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
360
90
360
90
Pentagon
5
540
108
360
72
Hexagon
6
720
120
360
60
Heptagon
7
900
128.6 (1dp)
360
51.4 (1dp)
Octagon
8
1080
135
360
45
Good luck with
your revision!