# octagon drawing help please



## sunnybob (Apr 3, 2015)

I'm going insane trying to draw an octagon on paper.
I have graph paper, compass, sharp pencil (I know, ancient history, but thats what I've got)

Try as I might, I cannot get all 8 sides an even length. I've even resorted to google images, but no result.

I want an octagon 68 mm across the flats if anyone can supply a printable image for me (g)


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## Semipro (Mar 22, 2013)

4 Ways to Make an Octagon - wikiHow


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## boogalee (Nov 24, 2010)

https://en.wikipedia.org/wiki/File:OctagonConstructionAni.gif


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## neville9999 (Jul 22, 2010)

Draw a Circle with a compass and then don't change the compasses arc, then put the point anywhere on the circumference of the circle and use the pencil end to mark a line around the curve, mark it in both directions, then put the point on one of the marks and mark it again in both directions, do that all the way around the circumference of the circle and then join the dots with straight lines, you will have a perfect Octagon, then you can see what the side of your Octagon is with that radius, change the radius until you get 68mm between the marks. N


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## tomp913 (Mar 7, 2014)

Draw the X & Y axes. From the intersection, and using the compass, draw a circle with 34 mm radius. Draw lines parallel to the X & Y axes and tangent to the circle. From the intersection of the axes, draw lines at 45° to cross the circle. Where these lines intersect the circle, draw lines perpendicular to them and tangent to the circle.


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## The Hobbyist (Apr 25, 2015)

Geometry was my favorite subject in school. My geometry teacher explained the importance of *measuring an angle correctly* in this way:

One of the odd things about geometry is circles, triangles, squares, hexagons and octagons.

A circle is exactly 360 degrees. If you draw a straight line from the center point of the circle to the circle itself, the line that forms the circle will give you exactly 360 degrees of rotation when you follow it around from where the line intersects the circle back to where it intersects the circle.

If you draw a line from the center point of a square to any corner, you will move exactly 360 degrees by following the line from that point, around the square to that same point. The same can be said for a triangle, a hexagon, an octagon, or any other closed shape.

A square has four equal sides, and when each corner is exactly 90 degrees, you will have a perfect square. (4) corners x 90 degrees of turn in each corner = 360 degrees.

A triangle has three equal sides, and when each corner is exactly 60 degrees, you will have a perfect triangle. (3) corners x 60 degrees of turn in each corner = 180 degrees

A Hexagon has six equal sides, and when each corner is exactly 120 degrees, you have a perfect hexagon. (6) x 120 degrees in each corner = 720 degrees.

An Octagon has eight equal sides, and when each corner is exactly 135 degrees, you have a perfect octagon. (8) x 135 degrees in each corner = 1,080 degrees.

How is that possible, and why don't the angles around a triangle, a square, a hexagon and an octagon all total 360 degrees, just like the circle?

A square, a triangle, a hexagon and an octagon all have a single line that begins at one point and moves around the shape until it returns to that point. Since we know that a full circle is 360 degrees, why 
does a square, a triangle, a hexagon or an octagon measured in the same manner give is different totals on the angles we measure? It is because an angle in a closed shape must be measured from the OUTside of the shape, NOT the INside. When two straight lines intersect, we are given TWO angles to measure, but one angle can be determines by subtracting the other angle from 180 degrees.

So a triangle has three 60 degrees angles, but we measure the angle from the OUTside. (3) x 120 degrees = 360 degrees (The direction of the straight line has been diverted by 120 degrees.)
A square has four 90 degree angles. Measuring from the outside gives us (4) x 90 degrees = 360 degrees (The direction of the straight line has been diverted by 90 degrees.)
A hexagon has six 120 degree angles. Measuring from the outside gives us (6) x 60 degrees = 360 degrees (The direction of the straight line has been diverted by 60 degrees.)
An octagon has eight 135 degree angles. Measuring from the outside gives us (8) x 45 degrees = 360 degrees (The direction of the straight line has been diverted by 45 degrees.)

Therefore, a square is NOT measured by determining that the inside angle is 90 degrees, but by determining that the OUTside angle is 90 degrees, meaning that the straight line has been diverted from its original direction by 90 degrees.

Does that clear things up? :nerd:


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## Nickp (Dec 4, 2012)

Using a chord length of 68mm, find the radius. Do this by using sine (opposite / hypotenuse) solving for 22.5 degrees. This is half the angle of 45 deg which would create a polygon. At this point you are solving for the hypotenuse of 34mm base (opposite side of right triangle made from isosceles triangle) which will become the radius. Once you've found the radius you can draw the circle with your compass then open your compass to 68mm and create equal points around the circumference. 

...make sure your pencil is sharpened... 

EDIT...should have followed all the way through... 34 divided by sine 22.5 = radius


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## tomp913 (Mar 7, 2014)

This is how we used to lay out octagon tables

- cut the square panel to the size over flats
- draw the diagonals and then the circle from the intersection of the diagonals
- draw the 45° across the corners where the diagonals intersect with the circle

HTH


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## sunnybob (Apr 3, 2015)

I am certain that its my human error, but I have tried most of these methods, and the result is quite a lot out. One of the shapes, despite measuring every angle twice, left me with two sides more than 2 mm longer than the others.
Doesnt sound much, but when its only 68 mm across, it looks bloody awful.

Another one, despite being measured to within the tolerance of the pencil line, looked like the top flat was 4mm further to the right than the bottom flat.

Why is something so simple, so hard? By the time I add the human tolerance of the mitre saw cut, its scrap.
I'm going to try the folded paper method later, see what transpires.
Can anyone supply octagonal stock?


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## Bushwhacker (Jun 16, 2009)

Each section should be 22.5 degrees. Draw the circle, split it into quarters, then split the quarters.


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## sunnybob (Apr 3, 2015)

Bushwhacker said:


> Each section should be 22.5 degrees. Draw the circle, split it into quarters, then split the quarters.


divide quarters by quarters and you get sixteenths.

45 degrees gets you an octagon, my problem is getting a REGULAR octagon. All eight sides an equal length


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## PhilBa (Sep 25, 2014)

Bob, it's not very clear what you are doing wrong, hopefully this will help.

I've attached a picture using 18mm stock. 

Get the angle right - Tilt your TS blade to 22.5 degrees. You should run 2 test pieces through and check to see that when joined they are 45 degrees. Adjust accordingly. Do one more and that should give you a 90 degree angle. To be absolutely certain, do all 8 test pieces and tweak as necessary.

Get the width right - Assuming your TS is left tilt, set your fence so that at the TOP of the wood (at 18mm or what ever you are using) the blade exits the wood exactly 68mm from the fence. You need to measure that distance - don't use the saw fence's ruler. It might require a little trial and error. 

Once you have the width right, rip and repeat.

good luck.

As an alternative, you can get 22.5 degree bevel bits for the router. The Freud 40-101 is one example but it's imperial, don't know if there's a metric one. But why do that? It's too easy and takes all the fun out of the screwing around with your table saw...


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## PhilBa (Sep 25, 2014)

Here's a printable picture.


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## tomp913 (Mar 7, 2014)

tomp913 said:


> This is how we used to lay out octagon tables
> 
> - cut the square panel to the size over flats
> - draw the diagonals and then the circle from the intersection of the diagonals
> ...


Now I'm confused. I thought you were looking for an octagon 68 mm between opposite parallel faces. If so, I think this is it - took me a while to figure the settings for the printer to give a full-size print, hopefully that's what you get on the other side. There's probably a way for me to specify metric dimensions, I just converted 34 mm to 1.3386". Not sure why there's .0001" difference in the side lengths, but it seems to be consistent.


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## ThomL (Oct 1, 2012)

Try the web site Calculators & Templates - Builders, Carpenters, Woodworkers & DIY it has many calculators for all kinds of things. Once in blocklayer click on Triangles - circles then polygons. You can change the radius length of your octagon and it will calculate the chord length for you. The diagrams can be Metric or Imperial and are printable to scale. 

It's a great site, I used it for gambrel roof rafters on my shed. The blocklayer site also has templates for dovetails, scalable fonts for signs and many other fun things. 

Have fun,
Tom


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## DaninVan (Jan 1, 2012)

Thanks, Tom; that's an excellent site. I love the pulley calculator...allows you to play with the input factors to home in on a desired result.


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## Router Roman (Jun 7, 2012)

Regular Polygon Calculator
Does this help at all?
Roman Zubar


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## sunnybob (Apr 3, 2015)

you all know that I'm pretty crap at woodwork, and I'm just playing as a retirement hobby, yeah?

That said, i was making a small trinket box for a grand daughter,
i layered 5 pieces of mahogany, all 6mm thick, at 90 degrees cross grain each layer.
So i have a square (cube?) sandwich of wood 70 mm wide x 30mm deep.
I make this stuff up as I go along (you can tell, cant you). Then I drill a 50mm hole through the centre of it using a hole cutter to make a band saw box. Top and bottom in a different wood to be added later.
Now I want to make the external shape an octagon, so i'm trying to draw the shape on paper, to glue it to the box, to cut the correct shape.
Sounds great on paper, but I'm in my second sandwich, and I can NOT get a true octagon template.
the cross grains look great on the edges, but its so obvious that the sides arent equal that the piece is scrap.

I dont have a good table saw, I'm cutting using the mitre saw from the paper template. 
I'm many hours ahead of most of you, so its evening here now, I shall try a few more of your suggestions tomorrow.

keep 'em coming


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## Nickp (Dec 4, 2012)

Bob...it seems that the key to your drawing is the radius of the circle you need. I assume you want each of the sides of the poly to be 68mm...?

Your radius should be 88.8463mm...sorry to be so precise...choose your own rounding... 

So if you draw a circle with that radius you can then extend your dividers to 68mm and pick any point on the circumference to start making marks on the circumference. Place the needle on each subsequent mark...

The radius I gave you is based on the 68mm being the chord of the arc created by a 45 deg angle (makes a polygon).

Are you having an issue with construction or just the drawing...? It seems @Phil has drawn a "constructable" diagram...

...BTW...Hipparchus and Ptolemy would be proud of all of us...  

Who'da thunk algebra, trig and geometry could be so much fun...    I'da listened more in school if I thought I would ever use it...


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## sunnybob (Apr 3, 2015)

Nick, you seem to have grasped the problem, but how the hell do I set my compass to 88.8463? And then hold that accuracy on the pencil point?

I tried at first to just cut 45 degrees, and then 45 degrees, and then...etc etc. The thing looked like it part melted and swayed 10mm to the right. There was a 4mm discrepancy between shortest and widest faces.

Thats when the paper template idea took hold, but it just isnt accurate enough.
My problem is (as I've said) my machinery is not 100% accurate. My mitre saw is pretty close now, after modifying the swivel detent 2 degrees to the right, and making zero clearance inserts, and ignoring the built in laser, but the compound errors just keep compounding.
I think I might have to drop the 8, and go for the much easier 4 sided.
but even my wife said how much she liked the alternating grain edges, so i would like to succeed just once.

Algebra Fun?, not really for me, I'm a point and shoot kind of guy, hate it when i have to use my brain.


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## sunnybob (Apr 3, 2015)

Phil, story covered as per above, thanks for the diagrams, but I'm working with solid wood, trying to cut the outside from a square to an octagon.
i think I'm working above my pay grade here, i might have to lower my sights a bit.


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## Stick486 (Jan 4, 2013)

another...


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## tomp913 (Mar 7, 2014)

Nickp said:


> Bob...it seems that the key to your drawing is the radius of the circle you need. I assume you want each of the sides of the poly to be 68mm...?
> 
> Your radius should be 88.8463mm...sorry to be so precise...choose your own rounding...
> 
> ...


If the material is 70 mm square x 30 mm thick, a radius of 88.8463 mm isn't going to work.


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## tomp913 (Mar 7, 2014)

Stick486 said:


> another...


Right, but I think that he's asking for A to be 68 mm. We made many octagonal tables using the method I described, but a 46-1/2" octagon (48" less 3/4" edge banding) is a lot more forgiving than something 2-5/8" across the flats. 

Laying out the octagon on his material is pretty straight-forward, starting with the material cut to 68 mm square, do the diagonals to get the center, draw the diagonals and find the intersection of the radius and diagonal and then use a combination square to mark the clip on each corner. But it's not going to work well if the saw isn't cutting square in both planes.


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## Stick486 (Jan 4, 2013)

tomp913 said:


> Right, but I think that he's asking for A to be 68 mm. We made many octagonal tables using the method I described, but a 46-1/2" octagon (48" less 3/4" edge banding) is a lot more forgiving than something 2-5/8" across the flats.
> 
> Laying out the octagon on his material is pretty straight-forward, starting with the material cut to 68 mm square, do the diagonals to get the center, draw the diagonals and find the intersection of the radius and diagonal and then use a combination square to mark the clip on each corner. But it's not going to work well if the saw isn't cutting square in both planes.


if *''A''* is 68MM then* ''C''* = 17MM and *''B'' *= 34MM...

if the saw isn't cutting square or the an accurate desired angle it's time for a tune up on the saw(s)...


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## Nickp (Dec 4, 2012)

You are so right, Tom...I was still working with the original post and missed Bob's further explanation. As I think of it now I assumed he was looking for a chord length of 68mm...it's obvious now that he meant 68mm from flat to flat...missed that completely.

...fun exercise though... 

Thanks for the correction...


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## Ghidrah (Oct 21, 2008)

This is the method I use, it works extremely well. 2/45° squares would make it simpler but 1 is fine.

1. I set and draw the Horizontal axis 1st and then poke a hole at, e.g., 6" then set the Vertical and Diagonal axis's. 
2. I then decide whether to use the inscribed or circumscribed diameter of the octo and mark it from the hole made in step 1.
3. I then connect the H, V and D facet points to complete the octo.

The only facet length errors made relate to your own accuracy when setting the squares and marking facet points


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## tomp913 (Mar 7, 2014)

Stick486 said:


> if *''A''* is 68MM then *''C''* = 24MM and *''B''* = 48MM...
> 
> if the saw isn't cutting square or the an accurate desired angle it's time for a tune up on the saw(s)...


I get B = 28.166 mm (1.109"), and C = 19.916 mm (0.7841") by calculation. Per your sketch, A = B + 2*C, but the above numbers add up to 96 mm.


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## Stick486 (Jan 4, 2013)

tomp913 said:


> I get B = 28.166 mm (1.109"), and C = 19.916 mm (0.7841") by calculation. Per your sketch, A = B + 2*C, but the above numbers add up to 96 mm.


bad math....
thanks for catching that...

as reads...

if *''A''* is 68MM then* ''C''* = 24MM and *''B'' *= 48MM...

amended to read...

if *''A''* is 68MM then* ''C''* = 17MM and *''B'' *= 34MM...


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## Nickp (Dec 4, 2012)

DaninVan said:


> Thanks, Tom; that's an excellent site. I love the pulley calculator...allows you to play with the input factors to home in on a desired result.


 @DaninVan This is a neat site...used it many times before. Nice part is it prints the plans for you when you're done. Makes for a nice document for the "project scrapbook"...


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## tomp913 (Mar 7, 2014)

Now that the ball game is over, I took a few minutes to learn how to change units in this CAD program. The attached is the layout of the octagon, dimensioned in millimeters. I printed it before realizing that the "R" dimension was missing - this should be 36.8013 mm.

The file prints "to scale" on my printer, not sure what it will do once it's been turned into a .pdf file.


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## jw2170 (Jan 24, 2008)

Tom's explanation combined with Stick's graphic gets my vote....


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## DesertRatTom (Jul 3, 2012)

An octagon is formed by drawing two squares. Draw the first 68 mm wide/tall. Bisect all 4 sides by putting a mark in the middle of each side. Connect the marks. Erase the resulting points.

Check that the connecting lines are at 45 degrees to the vertical and horizontal lines of the square. Should be right on.

You could also do this by cutting a perfectly square piece of paper or cardboard and using it to align the first square and the overlay square. Draw lines using the template. Mark the center point of each side on the template, line those points to align the diagonal outline to the first square Should be at a precise 45 degrees to the first square. Use an precise draftsman's right triangle.

Whichever way you draw the first square, make sure the baseline is perfectly parallel to the horizontal lines on the graph paper. use a horizontal line on the paper to make certain you have the a perfect 45 degree angle on the second square.

Even with a very fine pencil tip, getting the marks at exactly 34 mm to be exact will likely produce a slignt error, but that's a function of the pencil line's thickness, I'd be happy with 1/8 mm or less of an error. 

Hope that helps, been thinking about your question most of the day. BTW, this was drawn in Msoft Word, so it is not exact., just meant to give you the idea.


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## tomp913 (Mar 7, 2014)

View attachment octagon_metric.pdf
Exactly, two squares, one rotated and superimposed. If you clean up the extraneous lines on my layout, you can see that. A little hard to lay out on the part though.


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## tomp913 (Mar 7, 2014)

Stick486 said:


> bad math....
> thanks for catching that...
> 
> as reads...
> ...


I think the length of the side of an octagon is 2*(tan 22.5° x radius of the inscribed circle) or B = 2*(.4142 * 34) = 28.1665 mm


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## DesertRatTom (Jul 3, 2012)

tomp913 said:


> I think the length of the side of an octagon is 2*(tan 22.5° x radius of the inscribed circle) or B = 2*(.4142 * 34) = 28.1665 mm


If the squares are cut to 68mm on each side, then the size will be correct.


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## papasombre (Sep 22, 2011)

neville9999 said:


> Draw a Circle with a compass and then don't change the compasses arc, then put the point anywhere on the circumference of the circle and use the pencil end to mark a line around the curve, mark it in both directions, then put the point on one of the marks and mark it again in both directions, do that all the way around the circumference of the circle and then join the dots with straight lines, you will have a perfect Octagon, then you can see what the side of your Octagon is with that radius, change the radius until you get 68mm between the marks. N


Hi, Neville.

Unless you have a new drawing rule, it looks to me the way to get a perfect hexagon.


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## mjadams61 (Dec 24, 2015)

Here is something similar with what Tom was showing you but I always done octagons this way so hopefully it will help you.

You can measure and mark what size you want the 14 1/16 is just gives you a general Ideal. What I find that helps is finding or making your own centering ruler.It is easy to make a centering ruler out of metric or standard.


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## Ghidrah (Oct 21, 2008)

Speak of the devil, or enter whatever the correct phrase should be here ->( ) I remembered another way to do a near perfect octo last night while working out a new picnic table top.

All it takes is a pencil, ruler and compass.

1. Locate the Hor/Vert Ø of the sheet of paper and mark it. If writing paper use the line closest to Ø, to strike a line across then mark and strike the Vert line. 

2. Set your compass for circle to desired scale and mark it.

3. Position compass point where Hor and Vert lines intersect circle then scribe "Xs"

4. Strike lines to onnect the "Xs"


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