Mr F

1
Mr Fs Maths Notes

• Shape and Space
• 2. Polygons

2
2. Polygons
One of Mr Fs Top 1010 Maths Jokes What did the
pirate (who was also a very keen mathematician)
say when his parrot flew away?... Poly-gon!
you cant 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 Irregula
r Polygons Youve 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
12 sides
10 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 180
Why? Well, its 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 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
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 4 360 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
10

• Good luck with your revision!