### Constructing a Regular Decagon by Cutting the Corners off
a Regular Pentagon

To construct a regular decagon by cutting off the corners of a
regular pentagon,

the question is what is x? If the decagon is regular then we have

Where

is the proportion of the golden mean.Transpose the 2x to the other
side of the equation

Rationalize the denominator.

which reduces to

or

or

While this format doesn't have rationalized denominators, it is a
constructible number. One way to construct the regular decagon by
cutting off the corners of a regular pentagon is as follows.

In the figure, ABC is a square. E is the midpoint of BC. As a
result, we can use the Pythagorean theorem

to find that

F is where the unit circle centered at A meets AE. G is the foot
of F in AB.

By a simple ratio,

we get that

We now return to our earlier figure

to see that

If we let I be the midpoint of HF then

which is the amount we have to take off the corners of the
pentagon to get a regular decagon.

If you compare the construction of the regular
octagon from the square and the regular
dodecagon from the regular hexagon, this
one is much more complicated. It is not clear if there even is a
method for verifying this construction geometrically as there is for
the octagon and
dodecagon. It also casts doubt on our
quest to find a generic method which will work for all regular
polygons. However, there is a generic method which will work for all
regular polygons which, while it isn't as interesting as our methods
for the octagon and
dodecagon, is simpler in the case of the
decagon.