How to construct a pentagon

[mklotz]

In a recent thread (which I don't want to hijack further), I outlined a simple method to manually (i.e., no indexer, no CAD) divide a circle into 3,6,12,24,48... divisions.

It occurred to me that many folks may be interested in ten divisions. Ten divisions are trivial if one can do five divisions. So the question becomes, "How to construct a pentagon with only straightedge and compass?" - a so-called Euclidian construction.

[Aside: Not all regular polygons have a Euclidian construction. Gauss proved that certain ones (e.g., the heptagon - with seven sides) absolutely cannot be so constructed. Fortunately, the pentagon does not fall into this forbidden group.]

For most modelmakers there will be easier methods available to subdivide a circle. Still, it's nice to know how our craftsmen forefathers managed in the days before indexers and computers. And there will be times when the job is too big or awkward to fit into more conventional indexing devices.

The diagram below should be self explanatory. If you want to verify the technique, you can use trigonometry to prove that the angle AOE is indeed 360/5 = 72 deg.

 

[Andrewinpopayan]

The angle ACD remains constant?

 

[mklotz]

The angle ACD remains constant?

Yes, it does, as can be seen from the following...

tan (OAC) = (r/2)/r = 1/2 (where r = circle radius)

therefore: OAC = 26.565

OCA = 90 - OAC = 63.435

ACD = OCA/2 = 31.717

 

[arnoldb]

Thanks Marv

This brought back some nice memories from the draughting class from my high-school days

We used this same principle for constructing pentagons for drawings; - I used to love doing the technical drawings; some of the only homework I ever did ;D

I use glorified calculators (computers) extensively, but I have on occasion felt like breaking out my old draughting set, ink pens & just going for it

Regards, Arnold

PS: the spell checker keeps saying draughting is not a word, so I just shoved the word into Google and got only South African results - it seems the more acceptable word is "drafting"

 

[John S]

Depends if you have the door open or not when typing............... :

 

[arnoldb]

Depends if you have the door open or not when typing............... Roll Eyes

:big: Too true - guess I'll have to start closing doors & windows by 72 degrees; winter is coming on in Windhoek ;D

 

[Andrewinpopayan]

How do I bisect angle OCA without a protractor or am I missing something?

 

[djc]

How do I bisect angle OCA without a protractor?

You need a pair of compasses. With C as centre, strike an arc (of any radius) that intersects OC at X and CA at Y. Now with X and Y as centres, strike (equal radius) arcs that intersect at Z. Join Z to C and project if necessary to intersect AO at D.

http://strader.cehd.tamu.edu/geometry/bisectangle1.0/bisectangle.html

 

[mklotz]

Arnold,

In a language where 'fish' can be spelled 'ghoti', draught is perfectly acceptable. Don't listen to John. He's still working on the difference between 'course' and 'coarse'.

 

[tel]

Now that's an oddity - perhaps the influence of the US on 'English'?

Over here it's always been 'draught' and/or 'draughtsman'.

Here's a current definition

 

[John S]

Arnold,

In a language where 'fish' can be spelled 'ghoti', draught is perfectly acceptable. Don't listen to John. He's still working on the difference between 'course' and 'coarse'.

Of coarse or should that be course ????????

 

[tel]

PS: the spell checker keeps saying draughting is not a word, so I just shoved the word into Google and got only South African results - it seems the more acceptable word is "drafting"

.... or you can do what I just did - right click on the underlined word and hit 'add to dictionary'

 

[tel]

Of coarse or should that be course ????????

Coises! Foiled Again!

 

[Mainer]

The old (ca. 1806) portion of the house I used to own had a 5-sided ridge beam, which gave a roof pitch of about 8.7:12. The carpenters undoubtedly knew their geometry.

 

[Andrewinpopayan]

You need a pair of compasses. With C as centre, strike an arc (of any radius) that intersects OC at X and CA at Y. Now with X and Y as centres, strike (equal radius) arcs that intersect at Z. Join Z to C and project if necessary to intersect AO at D.

http://strader.cehd.tamu.edu/geometry/bisectangle1.0/bisectangle.html

perfect!

 

[mklotz]

Andrew,

If your memory of Euclidean construction is rusty, you might want to visit

http://whistleralley.com/construction/reference.htm

for a review of some of the fundamentals.