Angles and Polygons
By Mr Robertson
In and Out
Exterior angle
In this pentagon there are:
• 5 Interior angles
Interior angle
• 5 exterior angles
• 5 sides
Exterior Angles
The sum of the exterior angles = 360
Another way to express this is to say that the
number of exterior angles × the size of each
exterior angle = 360
And since the number of sides is the same as the
number of angles we can write:
number of sides x exterior angle = 360
se = 360
Exterior Angles
And if we know the size of the exterior angle we
can find the number of sides:
number of sides = 360 ÷ exterior angle
s = 360
e
Exterior Angles
We can rearrange this so if we know the number
of sides we can find the exterior angle:
exterior angle = 360 ÷ number of sides
e = 360
s
In and Out
Look at the diagram and you will see that the
exterior and the interior angle combine to make a
straight line.
This means that:
interior angle = 180 – exterior angle
exterior angle = 180 – interior angle
i = 180 – e
e = 180 – i
Summing Up
Sometimes you have to calculate the sum of the interior angles. This is going to be equal
to the number of sides × the size of the interior angle. This can be written like this:
Sum of interior angles = si
But the interior angle is just 180 – the exterior angle, so we can write it like this instead:
si = s(180 – e)
But the exterior angle is just 360 ÷ the number of sides, so we can write it like this instead:
si = s(180 – 360 ÷ s)
And if we multiply out this bracketed expression and simplify we get:
si = 180s – 360
So it turns out we can work out the sum of the interior angles, just by knowing the sides!
In and Out
• So it turns out that we can work out
everything else just by knowing one of:
– Number of sides
– Size of exterior angle
– Size of interior angle
– Sum of interior angles
• Let’s see an example
Example
• The sum of the interior angles of a regular polygon is 9000°. How
many sides has it got?
• Start with:
si = 9000
• Then replace i with 180 – e to get:
interior angle = 180 – exterior angle
s(180 – e) = 9000
• Then replace e with 360/s to get:
s(180 – 360/s) = 9000
exterior angle = 360 ÷ number of sides
• Multiply out the bracket and solve:
180s – 360 = 9000
180s = 9360
s = 9360 ÷ 180
s = 52
• So the polygon has 52 sides