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Octagon Math- Calculate the length of one side

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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
    "Objects in History may be closer than they appear"

  • #2
    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; 03-11-2008, 02:04 PM.
    "I cannot teach anyone anything; I can only make them think." - Socrates

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    • #3
      Re: Octagon Math- Calculate the length of one side

      Originally posted by JShade View Post
      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.



      John
      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 16-11/16" (Side Length)

      40 x .414214 = 16.56854" or 16-9/16" (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" or79-1/2" (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

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      • #4
        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.
        "Objects in History may be closer than they appear"

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        • #5
          Re: Octagon Math- Calculate the length of one side

          Originally posted by JShade View Post
          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.
          Don't take me seriously, I was just kidding. Take a look in the Rough Framing forum and you'll see why I called it a "math forum" ...
          "I cannot teach anyone anything; I can only make them think." - Socrates

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          • #6
            Re: Octagon Math- Calculate the length of one side

            Originally posted by Joe Bartok View Post
            Don't take me seriously, I was just kidding. ....
            I've been reading a lot here before sticking my neck out and posting, so I'm quite prepared to get the asbestos fire suit out if necessary. :)

            All in all there's been some good info I've read, as well as some damn hilarious postings.
            "Objects in History may be closer than they appear"

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            • #7
              Re: Octagon Math- Calculate the length of one side

              Originally posted by Shade View Post
              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.
              John,

              No problem. There are many different ways to figure the sides out but that's about the fastest way.

              Just have to watch out for Joe B because he's the nasty guy on the forum always yelling at people and trying to kick people off the forum.......;-)
              Joe Carola

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              • #8
                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

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                • #9
                  Re: Octagon Math- Calculate the length of one side

                  Originally posted by Brian Crisler View Post
                  I have always used this:

                  Radius of octagon multiplied by .828

                  For example: 12' Octagon would be 72" x .828= 59 5/8"

                  Brian
                  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' 11-5/8". 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' 5-15/16".

                  Here's a quick drawing.
                  Joe Carola

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                  • #10
                    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

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                    • #11
                      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.

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