Associated Topics || Dr. Math Home || Search Dr. Math

### Constructing a Regular Pentagon

```
Date: 21 Feb 1995 21:18:47 -0500
From: Eric Robbins
Subject: Re: Pentagon

Greetings, from Bramalea Secondary School in Brampton, Ontario

We are interested in knowing how to construct a regular pentagon
using a compass and a straight edge.

Thanks for your time and help!

Eric Robbins
```

```
Date: 23 Feb 1995 16:58:24 -0500
From: Dr. Ken
Subject: Re: Pentagon

Hello there.

Annie Fetter from the Geometry Forum gave me this reference.  It's in
a book called "Geometry for the Classroom" by C. Herbert Clemens and
Michael A. Clemens.

We must start by designating a certain length as "one unit."
After that we must determine the length of Sqrt{5} - 1 units
(if you don't know how you'd do that, write back; that's a whole
other construction, but it's easier).

Start with a horizontal line, L.  Pick a point p on L that will be
the center of the pentagon.  At a distance of Sqrt{5} - 1 units
from p along L, mark point z.  Construct a perpendicular to L at z.
Next, construct a circle of radius 4 units about p.  The distance
from point y (the point where the circle intersects L at the right)
to point x (the point where the circle intersects the perpendicular
to L at the top) is the length of one side of the pentagon.  Place
the compass point on x and mark off this distance on the circle.
Then place the compass point on the point you just marked, and
mark off the distance again.  Repeat this [a total of] five times,
and then connect all of the points you have marked.  This is the
regular 5-gon, or pentagon.

So there you have it.  If you wanted to construct a regular pentagon
inscribed in a given circle, you could find the radius of the circle, and
call its length 4.  Then bisect it twice, and you've got a segment of
length 1, and you can proceed as directed.

I hope you have fun with it!

-Ken "Dr." Math
```

```
From: Cedric Boyd
Subject: Re: Needed construction to construct a pentagon.
Date: Tue, 6 May 1997

Hi my name is Cedric Boyd form Overland Park, Kansas. I am in
Geometery and I am doing a project making a dodecahedren. If you would
please explain to me how to find the length of Sqrt {5} - 1 units A.S.A.P..
If you have trouble remembering, it was the article in Ask Dr.Math on
February 21, 1995. Thank you.
Sincerely,
Cedric Boyd
cedricb@usa.net
```

```
Subject: Re: Needed construction to construct a pentagon.
Date: Tue, 6 May 1997

Hi there-

In my opinion, this is one of the cutest constructions out there.  First
we'll construct a segment of length Sqrt{5}, and then we'll subtract 1 from
it (mark off 1 unit on that segment, and what's left has length Sqrt{5} - 1).

Start with a segment AB whose length is 1, and construct a perpendicular
line to that segment, passing through B.  On this new line, mark off a
distance of 1 from point B, and call that new point C.  So now if you draw
segment AC, you'll note that triangle ABC is a 45-45-90 degree triangle, so
its hypotenuse has length Sqrt{2}.  Now do the same kind of thing you just
did - construct a segment CD of length 1, perpendicular to AC.  Then ACD is
a right triangle whose hypotenuse (by the Pythagorean theorem) has length
Sqrt{3}.

I bet by now you see where this is going.  You'll get the square root of 4,
then the square root of 5.  Note that this procedure can be used to get the
square root of _any_ integer.

So now that you've got a segment whose length is the Sqrt{5}, mark of a
distance of 1 on that segment, and what's left will have length Sqrt{5} - 1.

-Ken Williams
```
Associated Topics:
High School Constructions
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search