Here is a little.......
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.
If you have questions ask