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How
could we find the area of a regular pentagon?












OK, first
what is a "regular" pentagon? 


Is there
an irregular pentagon? 





A regular
pentagon means that all of the sides 


and
angles are the same as each other. 





So say
that we have this one ... 











What do
we do? 





The first
step is to cut up the pentagon 


kind of
like a pizza. 











Now cut
each of the pizza slices in half 


down the
middle. 











Those
last cuts go from the middle of each side 


to the
middle of the pentagon. 


When we
do that, 


we make
90 degree angles on each side. 


We also
cut the length of each side 


exactly
in half ... 











So now we
have a bunch of little right triangles. 


Ten of
them actually. 


Like
this: 











So now if
we were told the length of the other side, 


we would
be in business! 


Say we
are told it is about 4 (it's actually more like 4.1291). 


That
means the area of each triangle is about: 











Or if we want to be more exact: 











There are
10 triangles in the pentagon, so 











Or to be more exact: 











So to
solve the area of a regular pentagon, 


we need
to know two things. 





We need
to know the length of a side. 


We need
to know the length of a line 


from the
middle of a side to the middle of the pentagon. 





Math
types really like to make up special names for things. 


They made
up a name for that middle of the side 


to middle
of the shape line. 


They call
it the APOTHEM, (yeesh! what a name ...) 





Example: 


Find the
area of this pentagon ... 











copyright 2008 Bruce Kirkpatrick 
