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### Constructing Polygons

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Date: 06/03/98 at 10:26:43
From: Natalie Grant
Subject: Construction of polygons (decagon)

How do you construct a polygon in general?

Example: a decagon. I knew how to do it, but I can't remember, and
there are no book resources for this topic. So far I remember that
there are two ways to choose from in order to construct a polygon in
general. If you could please send me even one of these methods, I
would be really grateful.
```

```
Date: 06/03/98 at 14:31:43
From: Doctor Wilkinson
Subject: Re: Construction of polygons (decagon)

I'm assuming you mean a regular polygon, and that you are talking
about construction with a straightedge and compass. If you want to
construct just any old decagon (say), all you need to do is pick any
10 points on a circle and connect them up in order.

For regular polygons the problem is much more interesting, and it
turns out that in fact there is no general construction, so you may
not be remembering correctly. But the regular decagon can be
constructed with a straightedge and compass, so you remembered

The problem of which regular polygons can be constructed with a
straightedge and compass was solved by Gauss around 1800. He showed
that it is possible to construct a regular polygon with n sides if and
only if n is a power of 2 times a product of distinct prime numbers
each of which is one more than a power of 2.  For example, 3, 5, 17,
and 257 are all such primes, so it is possible to construct a regular
polygon with 3, 4, 6, 8, 10, 12, 15, 16, or 17 sides, but not with 7,
9, or 18 sides. The ancient Greeks knew how to construct regular
polygons with 5, 10, and 15 sides, but Gauss was the first to show how
to construct a regular polygon with 17 sides.

To construct a regular decagon, draw line segments from the center O
of the pentagon to two consecutive vertices A and B and bisect the
angle AOB.

To construct a line segment of length the square root of five, start
with a line segment of length 1, draw a perpendicular to it at one
end, and mark off a line segment of length 2. Then the hypotenuse will
have length the square root of five by the Pythagorean Theorem.

-Doctor Wilkinson,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```

```
Date: 06/03/98 at 15:23:40
From: Doctor Rob
Subject: Re: Construction of polygons (decagon)

Actually, you cannot construct just any regular n-sided polygon; n
must be of a very special form. Its prime factorization must be of
this form:

n = 2^e * 3^e0 * 5^e1 * 17^e2 * 257^e3 * 65537^e4

where n > 2, e >= 0, and e0, e1, e2, e3, and e4 are either 0 or 1.
This means that you cannot construct a regular heptagon (n = 7) using
compass and straightedge. The same can be said of n = 9, 11, 13, 14,
18, 19, and infinitely many other values of n.

The method to construct a regular decagon (n = 10) is to construct a
regular pentagon (n = 5) inscribed in a circle, and then construct the
perpendicular bisectors of the five sides (these are also the
bisectors of the five angles). The ten points where the five bisectors
intersect the circle are the ten vertices of the decagon. (Five of
them will also be vertices of the pentagon.)

To construct the regular pentagon, start with a circle with center O.
Draw a diameter AOB, and draw the perpendicular bisector COD. Find the
midpoint E of AO. With E as center, draw an arc with radius EC,
intersecting the segment OB at F. Set your compass with radius CF.
Now with center C, draw an arc intersecting circle O at G and J. With
center G, draw an arc intersecting circle O at C and H. With center J,
draw an arc intersecting circle O at C and I. Then connect C to G to H
to I to J to C with line segments. CGHIJ is the regular pentagon, and
side HI is parallel to the diameter AOB. If the radius of the circle
is r, then the side of the regular pentagon is r*sqrt[(5-sqrt[5])/2].

The constructions of the regular 17-, 257-, and 65537-gons are much
more complicated, but still possible.

For a table of regular polygons constructable with straightedge and
compass whose angles are whole numbers of degrees, see the Dr. Math
FAQ:

http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions
High School Geometry
High School History/Biography
High School Triangles and Other Polygons

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