
03112008, 01:27 PM #1Regular Contributor
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Octagon Math Calculate the length of one side
After doing several searches here and not finding anything, then sourcing back to my Building Trades handbook (From the International Correspondence School, Scranton, PA, circa 1942) for a formula for calculating the length of one side of a regular octagon, I thought I'd share the results for anyone else wondering if there is a simple formula or multiplier. This formula has also been verified via Autocad on several size octagons.
This is designed for regular octagons only, meaning width and height are equal in size. Multiply the overall length of one side x .41667, and this will net you the length of each side.
Helpful if necessary to calculate Fypon or other trim material for octagon windows, which of course I'd find the length and add enough to cover the mitre cuts of your trim material.
John

03112008, 01:59 PM #2Veteran Contributor
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Re: Octagon Math Calculate the length of one side
Maybe this thread should have been started in the math forum (a.k.a. "Rough Framing" forum). :)
Regular Polygon Calculator
Formulas and Diagrams
Hope this helps.Last edited by Joe Bartok; 03112008 at 02:04 PM.
"I cannot teach anyone anything; I can only make them think."  Socrates

03112008, 02:49 PM #3Veteran Contributor
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Re: Octagon Math Calculate the length of one side
John,
That number is close but it's not the right number..414214 is the correct number. It's the Tangent of 22.5°.
For small numbers it wouldn't really make a difference with framing. Trim however does make a difference.
I'll use a 40" wide as the length of the side.
40 x .41667 = 16.6668" or 1611/16" (Side Length)
40 x .414214 = 16.56854" or 169/16" (Side Length)
1/8" difference means alot on trim.
If we had a 16' wide Octagon.
192 x .41667 = 80.00064" or 80" (Side Length)
192 x .414214 = 79.529" or791/2" (Side Length)
That .41667 probably came from the Tangent of 5/12 which is 22.62° and that's for a 5/12 pitch. Some guys hold the square 5/12 to scribe an angle for an Octagon when it's really 4.970563 which is 22.5° for an Octagon.Joe Carola

03112008, 03:42 PM #4Regular Contributor
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Re: Octagon Math Calculate the length of one side
Thanks for the clarification Joe C.
Wasn't exactly sure where to put this Joe B, so building science sort of seemed like the right place.

03112008, 03:45 PM #5Veteran Contributor
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03112008, 04:14 PM #6Regular Contributor
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Re: Octagon Math Calculate the length of one side

03112008, 04:56 PM #7Veteran Contributor
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Re: Octagon Math Calculate the length of one side
Joe Carola

03112008, 06:17 PM #8Veteran Contributor
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Re: Octagon Math Calculate the length of one side
I have always used this:
Radius of octagon multiplied by .828
For example: 12' Octagon would be 72" x .828= 59 5/8"
Brian

03112008, 06:57 PM #9Veteran Contributor
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Re: Octagon Math Calculate the length of one side
Brian,
The radius from what I understand is running the direction that the hip would run in plan view. So the radius when scribed would be to the outside point of the hip and the octagon side walls would be on the inside of the circle, short of the circle.would be inside the circle.
If you measure the octagon from the outside of two of the 8 sides and it measures 12', half of that would be 6' and your side walls would be 4' 115/8". If you scribe that as your radius, the outside of the radius would hit the 12' side walls but fall short of the inside corners where the hip runs at 67.5°.
That measurement will be 6' 515/16".
Here's a quick drawing.Joe Carola

03112008, 09:41 PM #10Veteran Contributor
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Re: Octagon Math Calculate the length of one side
Joe:
Sorry, I thought he was just trying to determine the length of each parallel side of an octagon. The measurement that you gave for 12' is correct and is the measurement used to determine square of an octagon as well as rafter length. Sorry for any confusion that I may have caused.
Brian

08172013, 12:58 PM #11New Member
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Re: Octagon Math Calculate the length of one side
We laid down 12 inch pavers in the patio, leaving a few spaces to plant trees. to make it look nice i want to make those spaces octagonal, by adding back triangles in the corners.
it's the same problem as taking off the corners of a 12 inch tile to make an octagon.
from the value .4142 given above, the length of the side after cutting off two corners is .4142 x 12 = .497 inch. The cut will be between the points 3.51 inch either side of the corner.
That's very close to 5 inches and 3.5 inches, which is interesting because you referred to 5/12 "Some guys hold the square 5/12 to scribe an angle for an Octagon."
It's also interesting because 5 is the hypotenuse (long side) in the pythagorean theorem for a right triangle with sides 3, 4 and 5 inches. (3x3 + 4x4 = 5x5).
So if we take the average of 3 and 4 and make the 2 short sides each 3.5 inches, it's still pretty close.
that might be useful in trying to remember the factor.
Or else a ratio for a 24 inch octagon, 7+7+10=24 might be easy to remember. The side you cut off is 7/24 of the way from the corner.