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#### reible

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So you’re a woodworker are you? Just to keep this forum interesting for everyone I have a little problem for you.... but the good news is I also solve the problem and tell you how I did it. This uses MATH but it's OK you can do it!

The problem: I have a 2" dia. pipe I want to make an octagon to fit tightly around the pipe. For those who are trying to remember an octagon is like a US stop sign, 8 sides. Take a look at the attachment.

The angle for the cuts is done by taking 360 deg and dividing by the number of sides (8) then taking half that number for the angle of one segment. I know some of you are way ahead of me by now; yes the angle is 22-½ deg. So you go and cut the 8 pieces with 22-½ deg angles and …….. what length did you cut them?

You knew it was going along too easy now didn’t you. OK the math part is coming. For a moment let’s look at what we have, we have a polygon (the octagon) and it is “circumscribed” about a circle (drawn around a circle). We know the radius (half the dia. of the circle (1”), we know the number of sides (8) this lets us use a formula now don’t panic we are not really going to use this….. the perimeter, that is the lengths of all the inside surfaces of the octagon = 2nr(sq)r tan (pi/n) (sorry to you math people put I don’t know how to get the text to look like what it should) If we know what the perimeter is then if you divide that by 8 we know what the length of each of the pieces is. (This is the length of the short side of the piece facing the inside.)

So are we all lost yet? Well if we do the math it ends up being .828427” but some of you already knew that right???? Those people are all busy working at NASA or somewhere so I doubt any of us could have guessed this.

So how can I make this simpler you ask? While working out this math I found a short cut. I did a sample run of a lot of sizes and sides and it is always correct at least to the third decimal place which is fine for even those among us with fences that are good to .001”. For an octagon the length is equal to the radius divided by 1.2071. For our problem the radius was 1” so 1 divided by 1.2071 is = to .828431 or as I said .828”

Let’s say the pipe had been 8” then the radius would be 4” and the length we are looking for is 4 divided by 1.2071 or 3.313”.

For a triangle (3 sides) the number is .28867

For a square (4 sides) the number is .50000

For a pentagon (5 sides) the number is .68819

For a hexagon (6 sides) the number is .86602

For an octagon (8 sides) the number is 1.2071

For a dodecagon (12 sides) the number is 1.8660.

#### Maurice

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Gee Reible, I would do it this way:
Draw a two inch circle on paper, divide 360 degrees by 8 like you did,
draw a straight line touching the circle and extend it way out. Draw another line farther around the circle to meet the first line at 22 1/2 degrees, extend it well out.
Continue doing this till you have closed in the circle.
Now you can measure the length of the octagon lines where they meet.

There y' go
mo.

#### reible

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Maurice said:
Gee Reible, I would do it this way:
Draw a two inch circle on paper, divide 360 degrees by 8 like you did,
draw a straight line touching the circle and extend it way out. Draw another line farther around the circle to meet the first line at 22 1/2 degrees, extend it well out.
Continue doing this till you have closed in the circle.
Now you can measure the length of the octagon lines where they meet.

There y' go
mo.
I wish you would have said "This is they way I would have done it before I found out how easy it is to do your way."

Maybe if we see how long it takes each of us to do a sample problem might just have you "see the light". OK we have a planter that is 12" in Dia. and we want a nice cedar 12 sided box made to fit around it.

OK the radius is 6 so 6 divided by the 1.8660 is 3.215 ..... I'm done! I guess that was oh maybe 15 seconds..... please post your time when you finish........

Yes you can do a drawing, yes you can make it by guessing then trimming the parts and maybe another half dozen ways but this is a very fast way and a very easy way.

Now get out your calculator and give it a try!!!!!!!

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#### GoonMan

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508 Posts
roycoola said:
go to this site and you can figure any side of an octogon I make a lot of planters to I keep this site in my favorites http://buster2058.netfirms.com/octagon/calc/octagon_layout_calc.htm

Thanks roycoola,

That link helps in my not so good math mind. My wife wants me to make some planters out of Cedar Fencing and I have been puzzling over it off and on for a while on how I could do it.  #### Richard Y

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I like the Math answer, a lot easier than my solution. By the way anyone using the calculator at Buster would, according to thier drawing, get the wrong answer. thier drawing shows the radius to the apex of a joint, so the pipe would not fit inside.

Rick

#### fantasea

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It gets easier... At woodweb.com there's a program called polycut, it's freeware

The program calculates side angles, number of sides, in summary the time it took to write formula and calculate if this "free" program is used your already cutting wood... Time is money!!!
reible said:
So you’re a woodworker are you? Just to keep this forum interesting for everyone I have a little problem for you.... but the good news is I also solve the problem and tell you how I did it. This uses MATH but it's OK you can do it!

The problem: I have a 2" dia. pipe I want to make an octagon to fit tightly around the pipe. For those who are trying to remember an octagon is like a US stop sign, 8 sides. Take a look at the attachment.

The angle for the cuts is done by taking 360 deg and dividing by the number of sides (8) then taking half that number for the angle of one segment. I know some of you are way ahead of me by now; yes the angle is 22-½ deg. So you go and cut the 8 pieces with 22-½ deg angles and …….. what length did you cut them?

You knew it was going along too easy now didn’t you. OK the math part is coming. For a moment let’s look at what we have, we have a polygon (the octagon) and it is “circumscribed” about a circle (drawn around a circle). We know the radius (half the dia. of the circle (1”), we know the number of sides (8) this lets us use a formula now don’t panic we are not really going to use this….. the perimeter, that is the lengths of all the inside surfaces of the octagon = 2nr(sq)r tan (pi/n) (sorry to you math people put I don’t know how to get the text to look like what it should) If we know what the perimeter is then if you divide that by 8 we know what the length of each of the pieces is. (This is the length of the short side of the piece facing the inside.)

So are we all lost yet? Well if we do the math it ends up being .828427” but some of you already knew that right???? Those people are all busy working at NASA or somewhere so I doubt any of us could have guessed this.

So how can I make this simpler you ask? While working out this math I found a short cut. I did a sample run of a lot of sizes and sides and it is always correct at least to the third decimal place which is fine for even those among us with fences that are good to .001”. For an octagon the length is equal to the radius divided by 1.2071. For our problem the radius was 1” so 1 divided by 1.2071 is = to .828431 or as I said .828”

Let’s say the pipe had been 8” then the radius would be 4” and the length we are looking for is 4 divided by 1.2071 or 3.313”.

For a triangle (3 sides) the number is .28867

For a square (4 sides) the number is .50000

For a pentagon (5 sides) the number is .68819

For a hexagon (6 sides) the number is .86602

For an octagon (8 sides) the number is 1.2071

For a dodecagon (12 sides) the number is 1.8660.

#### [email protected]

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(1) Draw any size circle on a paper. That represents the OD of the pipe.
(2) Cut four strips of paper (any width) and lay/tape them at right angles to each other - so that they make a box around/touching the circle (tangents).
(3) Draw a line on each strip at the point of contact with the pipe. (These lines divide each strip in half.)
(4) Cut out four more identical strips and place/tape them together (like the first set).
(5) Now rotate the top square so that the corners are lined up with the midpoint lines on the first strips.
(6) Finally, cut off all protruding corners. ;~)

#### abacuscomp

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Here is an interesting concept for a hexagon of 6 sides:

Since the inside angels of all triangles must add up to 180 degs., and the hexagon is an equilateral triangle with each inside angel at 60 degs., and all three sides are the same length. Therefore, the inside cut length is equal to the length of the radius and the angle is 30 degs.

A = 60
R = 1.5
2X = 2 (r cos (a)) = 1.5 = R

Enjoy.

#### reible

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abacuscomp said:
Here is an interesting concept for a hexagon of 6 sides:

Since the inside angels of all triangles must add up to 180 degs., and the hexagon is an equilateral triangle with each inside angel at 60 degs., and all three sides are the same length. Therefore, the inside cut length is equal to the length of the radius and the angle is 30 degs.

A = 60
R = 1.5
2X = 2 (r cos (a)) = 1.5 = R

Enjoy.
Thanks!

For those that are getting lost here just remember this is for a circumscibed circle...... not an inscribed as I have shown......

Now remember this trick:

1*9 is 9
2*9 is 18 is 1+8 or 9
3*9 is 27 is 2+7 or 9
4*9 is 36 is 3+6 or 9
5*9 is 45 is 4+5 or 9
6*9 is 54 is 5+4 or 9
7*9 is 63 is 6+3 or 9
8*9 is 72 is 7+2 or 9
9*9 is 81 is 8+1 or 9
10*9 is 90 is 9+0 or 9
11*9 is 99 is 9+9 or 18
12*9 is 108 is 10+8 or 18
13*9 is 117 is 11+7 or 18
14*9 is 126 is 12+6 or 18
Who said math can't be fun!

Ed

#### Sidro

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How would I do it? I cheat and draw it on CAD. Sid

#### JamesEMc

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reible said:
Thanks!

For those that are getting lost here just remember this is for a circumscibed circle...... not an inscribed as I have shown......

Now remember this trick:

1*9 is 9
2*9 is 18 is 1+8 or 9
3*9 is 27 is 2+7 or 9
4*9 is 36 is 3+6 or 9
5*9 is 45 is 4+5 or 9
6*9 is 54 is 5+4 or 9
7*9 is 63 is 6+3 or 9
8*9 is 72 is 7+2 or 9
9*9 is 81 is 8+1 or 9
10*9 is 90 is 9+0 or 9
11*9 is 99 is 9+9 or 18
12*9 is 108 is 10+8 or 18
13*9 is 117 is 11+7 or 18
14*9 is 126 is 12+6 or 18
Who said math can't be fun!

Ed

Ed, you've got too much time on your hands.     Jim

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