But were Stymied by your inexperience with The Fundamentals of Geometry...??? ( ! )

It's easy enough, with only a few simple tools:

Such as;

A Sheet of Paper or

*Other Inscribable Surface,*

A StraightEdge;

*( Commonly called a Ruler ),*

A Compass

*( The Kind that Allows You to Draw*

*a Circle or Part of a Circle ( An Arc! ),*

A Protractor;

*( Typically--A Half-Circle of Plastic*

*with Marks to Indicate Angular Progress along The Arc )*

and a Writing or

*Inscribing Utensil.*

*To Draw an Eleven Sided Polygon*

*( An Ennagon )*

Simply divide The Number of Degrees in a Circle;

*( 360 )*

By The Number of Desired Sides

*( 11 )--*

Which Equals

*32.7273!*

Draw a Circle,

Carefully Marking where The Center is,

With a Radius of The Desired Amount,

Then Draw a Line through The Circle,

So that it passes through The Center,

This Line is to be Called

*'Alice'.*

Where this Line passes Through The Edge of the Circle,

Mark as

*'Betty'.*

Align your Protractor along

*Alice*

So that it's Center Marking Overlaps The Center of your Circle.

Then Find Where 32.7273 is along The Protractors Arc,

And make a small

*'Tick'*Mark on your Inscribing Surface.

Use your Straight Edge to Draw a Line

*( A Finite Ray )*

So that it passes through the Center of your Circle--

The Newly made

*Tick Mark*, and

*The Edge of the Circle.*

Where this Line Intersects the Circle, Label it

*'Cindy'*.

Carefully place

*The Pointy Metal Spike*of your Compass on

*Betty*,

And move The Adjustable Arm back & forth

Until it's

*Angle*reproduces The Length from

*Betty*to

*Cindy*.

*Take care to Preserve this Angle on your Compass...!!!*

...As you place The Pointy Metal Spike on

*Cindy*

And Make another tick mark along the Circle's Edge,

*Moving Away from Betty.*

*( You should be Moving Counter-Clockwise,*

*But if you're not, That's OK too...! )*

Repeat this procedure,

*over and over,*

Moving The Pointy Metal Spike at each interval

To the Tick Mark that you've just made,

Until your most recent

*Tick Mark*is where

*Betty*is.

It may be entirely possible

That this most recent

*Tick Mark*will not be exactly where

*Betty*is,

*But it should be pretty close...!!!*

**If it's WAY OFF...!!!**

Then you may either

*decide*that it's

*'Good Enough'*or you may want to

**.**

*Do It Over*When all the Tick Marks are Done;

Use your Straight Edge to Connect them into your

**Ennagon!**---

**Wasn't that Simple...?***But what if you Don't have a Protractor...!!!*

What if you have a Straight Edge, Compass, Writing Utensil & Inscribing Surface...

*But no way to determine how much of an Angle 32.7273 would be...???*

---

**A General Solution is one that applies**

*The General Solution!*

Not only to what you want to do for a Specific Problem,

*But will work on Any Problem of that Kind...!!!*

This Solution is

*Not Completely*a General Solution,

Since it will only work for Polygons up to

*19 Sides...*

*But that should be plenty,*

*Because--*

Even if you're

*VERY Neat & Careful*with A Straight Edge, Compass & Writing Utensil...

You really can't make Polygons with more sides than this...

*With any Reasonable Degree of Reliability...!!!*

---

**Step One:**

Draw a Circle of a The Desired Radius.

*The Radius is from Center to The Circle's Edge.*

*The Diameter is from The Circle's Side to Side.*

*When you're Calculating The Circumference of a Circle;*

*The Formula is 2 times Pi times r^2.*

*It's 2 times... Pi r squared,*

*Because Pi is a function of the Circumference, Not the Radius,*

*Even though all Modern Geometers always think of Circles*

*by their Radius' as how big they are...*

*The ancient Greeks, that Discovered pi,*

*thought about The Size of Circles by their Diameters!*

*But by the time Modern Geometry came along,*

*The Value of Pi was already firmly ingrained into everyone's heads...!*

**Step Two:**

Draw a Line through the Center,

Extending through both Sides, And then some.

**Step Three:**

Use this

**to find The Polygon that you're Interested in.**

*'Look Up Table'*Nearly Each Polygon is provided with 2 Right Triangles,

Such that the Accumulative Angles at The Center of the Circle

Will very nearly match the Desired Angle for Creating your Polygon... ( ? )

**To Use this Information:**

Find The Desired Polygon

Note The Angle,

*for your Personal Enrichment*

Then make a note of

**The Two Ratios**that Follow

The Triangles Angles are given after that,

*Again just for your Personal Enrichment,*

*What you're really;*

Along with the Margin of Error.

Along with the Margin of Error.

**And Only Interested in though,**

**Is The Ratios.**

*So; As an Example--*

For an

**Ennagon**;

The Two Triangles are

**7:3**&

**6:1**

These Triangles produce The Angles 23.2 Degrees & 9.46 Degrees,

Which Add Up to 32.66,

*Pretty close to the Desired 32.73*

All the Ratios are given with Fairly Small Integers,

So that the Triangles that you'll be drawing will be easy to measure... ( ! )

*Proceed to Step Four...*

Look Up Table for Polygon Drawing...

Name......Angle....Ratios......TriAngles........Error

Triangle..120......5:9 + 3:5...60.94 + 59.04....0184

Square.....90......1:1 + 1:1...45....+ 45.......0

Pentagon...72......5:6 + 5:2...50.19 + 21.80....0042

Hexagon....60......5:3 + 9:5...30.96 + 29.05....0184

Septagon...51.43...7:5 + 7:2...35.54 + 15.95....0545

Octagon....45......1:1.........45...............0

Nonagon....40......3:2 + 9:1...33.69 + 6.34.....0303

Decagon....36......2:1 + 6:1...26.57 + 9.46.....0274

Ennagon....32.73...7:3 + 6:1...23.2. + 9.46.....0664

12agon.....30......7:2 + 4:1...15.95 + 14.04....0184

13agon.....27.69...8:3 + 8:1...20.56 + 7.13.....0113

14agon.....25.71...3:1 + 8:1...18.43 + 7.13.....1543

15agon.....24......9:4.........23.96............5625

16agon.....22.5....5:1 + 5:1...11.31 + 11.31....1199

17agon.....21.18...8:1 + 4:1....7.13 + 14.04....0152

18agon.....20......9:2 + 8:1...12.53 + 7.13.....3462

19agon.....18.95...6:1 + 6:1....9.46 + 9.46.....0227

**Step Four:**

This Step Assumes

That you already know how to make a Right Angle

That Intersects with a given Line.

If you don't know how to do this with a Straight Edge & Compass,

*The use the Corner of a Sheet of Paper.*

**Then:**

You want to create a Triangle so that its bottom is The Base Line

That intersects through your Circle.

This Triangle also needs to have Sides that Equal the Ratio Values.

For this

*Ennagon*:

The first Triangle is

**7:3**...

That's 7 Units along The Bottom, and 3 Units Up.

These

*'Units'*may be

**,**

*any length*So long as they agree with one another.

You can even use different sized Units for the Second Triangle...!

*( Although... Why would you WANT to...??? )*

To get 7 Units,

Think of The Distance from The Center of your Circle

To the Outer Rim as 4 Units,

Then Divide this Length in Half, and Half again,

Yielding 1 Unit Increments.

The Double that 4 Unit Length Along the Base Line,

So that it's now 8 Units.

Subtract 1 Unit,

*So that it's 7 Units.*

Draw a Right Angle at this Point.

Extend that Height to 3 Units.

Then Complete the Triangle by Connecting

*That Point*to the Center.

This Newly Drawn Line is The New Base Line!

Repeat this Procedure for The Second Triangle. (

**6:1**)

*( You've already got your 1 Units, So this should be easy! )*

The Two Triangles now create a Accumulative Angle

That passes through your Circle at

*32.66 Degrees*.

Make a Tick along the Circles Edge

Where The Second Triangles

*Hypotenuse*Intersects it!

*( The Hypotenuse is the longest side of The Triangle! )*

Use this

*'Measurement'*with your Compass

To Mark Off the rest of Vertices ( Points ) for your Polygon.

Connect these Vertices with your StraightEdge!

**Wasn't that Easy...!**

*Your Done...!*

Well...

*Pretty Easy...???*

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