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Head First 2D Geometry by Dawn Griffiths, Stray

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regular polygons
Great tub choice!
Your pick of the six-sided hot tub has stirred up
excitement among all the bands booked for the
festival—they think it’s gonna be amazing. And you
didn’t even have to do any more math to choose the
hexagon tub, since you already found that handy trend.
Yeah—great, except that the
carpenter wants the dimensions and
angles of this fantasy six-sided tub
that you’ve picked. Please don’t tell me
that chopping it up into triangles is
going to find those as well....
T
he carp
enters
w
ant this
interna
l angle.
We know we can divide the hexagon tub into six congruent
triangles. From this, can you find:
1) The total of all the angles of those “sector” triangles.
2) The angle that the carpenters need to build the wood
frame for the hot tub: the internal angle of the polygon.
A handy
hexagon t
o
doodle on
Actually, it might.
It’s not a bad suggestion at all. We have plenty of triangle-
related tools in our toolbox, so when we’re faced with a
problem we can’t already solve, looking for ways to make it
about triangles is a great starting point. Let’s give it a go.
you are here 4 313
Number
of sides
All the
l
angle
wr
i
polygon internal angles formula
Angles in a triangle add up to 180º
There are six triangles so the total angles are:
6 x 180º = 1080º
The six angles in the middle add up to 360º
So the total of all the internal angles is the
total angles minus the six angles in the middle =
1080º - 360º = 720º
It’s a regular polygon, so the angles are equal = 720º = 120º
We know we can divide the hexagon tub into six congruent
triangles. From this, can you find:
1) The total of all the angles of those “sector” triangles.
2) The angle that the carpenters need: the internal angle of
the polygon.
Solution
6
1)
2)
Add up ALL
the angles in
the tr
iangles…
and then
subtrac
t these
six in the middle.
Internal angles of a polygon follow a pattern
A polygon can always be divided into the same number of triangular
sectors as it has sides. And every triangle has angles totaling 180º.
That’s the key to finding the internal angles of a polygon:
interna
l angles
-
=
Total angles in the Angles which aren’t
One interna
Total internal angles
triangles = part of the internal
= (n x 180
o
) - 360
o
180
o
x number of sides angles = 360
o
Each internal angle
= (n x 180
o
) - 360
o
T
his formul
a is also
tten as (n-2) x 180
n
n
314 Chapter 7

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