# Constructing a polygon within a given circle
**URL:** https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677
**Category:** Factual Questions
**Created:** [June 6, 2016, 9:49pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677 "2016-06-06T21:49:36Z")
**Posts on this page:** 14
**Page:** 1
### Author:  [@panache45](https://boards.straightdope.com/u/panache45)
#### Post date: [June 6, 2016, 9:49pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/1 "2016-06-06T21:49:36Z")
What is the regular polygon with the least number of sides that cannot be constructed within a given circle, using only a straightedge and a compass?
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### Author:  [@Nancarrow](https://boards.straightdope.com/u/Nancarrow)
#### Post date: [June 6, 2016, 9:52pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/2 "2016-06-06T21:52:30Z")
A heptagon (seven sides).
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### Author:  [@panache45](https://boards.straightdope.com/u/panache45)
#### Post date: [June 6, 2016, 9:59pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/3 "2016-06-06T21:59:47Z")
Yes, of course. I knew you could get close, but not precisely.
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### Author:  [@Giles](https://boards.straightdope.com/u/Giles)
#### Post date: [June 6, 2016, 10:20pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/4 "2016-06-06T22:20:48Z")
The complete list would start with:
7 sides
9 sides
11 sides
13 sides
14 sides
15 sides
18 sides
19 sides
21 sides
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### Author:  [@markn\_1](https://boards.straightdope.com/u/markn_1)
#### Post date: [June 6, 2016, 10:46pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/5 "2016-06-06T22:46:48Z")
See [Constructable Polygon](https://en.wikipedia.org/wiki/Constructible_polygon). The only regular polygons that can be constructed are those where the number of sides is a product of a number of distinct Fermat primes multiplied by a power of 2.
–Mark
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [June 7, 2016, 12:04am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/6 "2016-06-07T00:04:16Z")
15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
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### Author:  [@Giles](https://boards.straightdope.com/u/Giles)
#### Post date: [June 7, 2016, 12:57am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/7 "2016-06-07T00:57:18Z")
> [@Chronos](#):
>
> 15 sides is easily doable, since both 5 and 3 are constructible, and are relatively prime.
Of course you are right – my mistake.
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### Author:  [@Lumpy](https://boards.straightdope.com/u/Lumpy)
#### Post date: [June 7, 2016, 1:00am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/8 "2016-06-07T01:00:09Z")
Of course you can construct n-sided polygons with a _ruled_ straightedge; I wonder why the Greek geometrists considered that cheating.
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### Author:  [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)
#### Post date: [June 7, 2016, 1:15am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/9 "2016-06-07T01:15:12Z")
> [@Lumpy](#):
>
> Of course you can construct n-sided polygons with a _ruled_ straightedge; I wonder why the Greek geometrists considered that cheating.
Is there anything the Greeks cared about that _couldn’t_ be constructed with a compass and ruler? If there isn’t, that takes most of the game out of the game.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [June 7, 2016, 2:38am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/10 "2016-06-07T02:38:13Z")
You can make a ruled straightedge easily, using the Greeks’ own techniques (Euclid’s Proposition VI.9). The problem isn’t in putting the marks on the ruler; it’s in interpolating between the marks.
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### Author:  [@MikeS](https://boards.straightdope.com/u/MikeS)
#### Post date: [June 7, 2016, 2:53am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/11 "2016-06-07T02:53:00Z")
> [@Exapno\_Mapcase](#):
>
> Is there anything the Greeks cared about that _couldn’t_ be constructed with a compass and ruler? If there isn’t, that takes most of the game out of the game.
If [Wikipedia](https://en.wikipedia.org/wiki/Neusis_construction) is to be believed, you can [trisect an angle](http://www.cut-the-knot.org/pythagoras/archi.shtml) or double a cube using a marked straightedge. You cannot, however, [square the circle](https://en.wikipedia.org/wiki/Squaring_the_circle) (which is a problem the Greeks knew of and cared about.) Roughly speaking, doubling the cube becomes possible because the cube root of two is an algebraic number, while π is a transcendental (i.e., non-algebraic) number and so squaring the circle is still impossible.
There’s also a list of the regular n-gons that can and cannot be constructed using a marked straightedge [here.](https://en.wikipedia.org/wiki/Compass-and-straightedge_construction#Markable_rulers) In particular, it is still impossible to construct a regular [hendecagon.](https://en.wikipedia.org/wiki/Hendecagon)
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### Author:  [@Nava](https://boards.straightdope.com/u/Nava)
#### Post date: [June 7, 2016, 4:28am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/12 "2016-06-07T04:28:30Z")
> [@markn\_1](#):
>
> See [Constructable Polygon](https://en.wikipedia.org/wiki/Constructible_polygon). The only regular polygons that can be constructed are those where the number of sides is a product of a number of distinct Fermat primes multiplied by a power of 2.
>
> –Mark
I don’t know a Fermat prime from a hole in the wall, but would that “power of 2” include 2[sup]0[/sup]? Because otherwise I don’t know how you come up with 3, a number whose regular polygon can be drawn with only a compass.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [June 7, 2016, 10:41am UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/13 "2016-06-07T10:41:32Z")
Yup, 1 is a power of 2.
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### Author:  [@markn\_1](https://boards.straightdope.com/u/markn_1)
#### Post date: [June 7, 2016, 3:37pm UTC](https://boards.straightdope.com/t/constructing-a-polygon-within-a-given-circle/756677/14 "2016-06-07T15:37:48Z")
Yes, as Chronos says, 1 is a power of 2.
Also, “a number of Fermat primes” includes the case of zero Fermat primes, which is why you can construct a square (4 sides = 2^2, with no primes involved).
A Fermat prime is a prime of the form (2^(2^n))+1. There are only 5 known Fermat primes: 3, 5, 17, 257, and 65537.
–Mark