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Thread: Math Quandry-

  1. #1
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    Math Quandry-

    I’m in a bit of a math quandary.

    I know that if I cut 8 pieces of identical size stock with an 11.25 degree bevel on each side I can assemble them into a perfect half-circle. What I can’t figure out though is how to determine the width of the pieces in order to come up with a pre-determined radius.

    I’ll bet someone can embarrass me by pointing out something I should have learned in the sixth grade!

    Thanks,


    Rick
    I'm only responsible for what I say, not for how it is understood

  2. #2
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    You can always use trig (which, no, you probably did not learn in 6th grade) to figure this out -

    sin 22.5 * radius of circle = outside length of each piece...

    assuming this was the desired outcome:
    http://images.rockler.com/rockler/images/24784-01-200.jpg
    Last edited by Matt Armstrong; 06-18-2009 at 1:05 PM.

  3. #3
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    I also have to point out the irony of our avatar juxtaposition

  4. #4

    Radius is half the length of the Diameter.

    Diameter is the measurement from one side of a circle to the other directly through the widest part (middle). The Radius is half of the length of the Diameter.

    So if you wanted to create a 20" Diameter circle, the length of each piece would be half of that at 10" which is the Radius.

    Similarly, if you wanted to end up with a 30" Diameter (15" radius), the length of each piece would be 15".

  5. #5
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    Quote Originally Posted by Grant Morris View Post
    Diameter is the measurement from one side of a circle to the other directly through the widest part (middle). The Radius is half of the length of the Diameter.

    So if you wanted to create a 20" Diameter circle, the length of each piece would be half of that at 10" which is the Radius.

    Similarly, if you wanted to end up with a 30" Diameter (15" radius), the length of each piece would be 15".
    Actually Grant, that is not what Rick is after. He wants to know how wide his 8 boards need to be to hit a given half of a circumference.

    So since the circumference of a circle is 2 * Pi * radius, half of that is clearly Pi * radius.

    So what Rick is after for a given radius is

    8 * width = Pi * radius

    Since radius is known ahead of time.

    width = (Pi * radius) / 8

    Cheers

    Brian
    Last edited by Brian Willan; 06-18-2009 at 1:16 PM.

  6. #6
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    brian's way is even easier... d'oh!

  7. #7
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    Rick,
    For starters 8 pieces with an 11.25 degree bevel will only give you 1/4 of a circle, not 1/2. I am assuming that when you talk about length of the pieces, you mean the distance from point to point after you assemble the 8 pieces. If this is the case, then the length or chord length = 2 x Radius x Sin(A/2) where A is the 11.25 degree angle. For example, if you wanted the outside radius of your circle to have a 40" radius, the length of the long side of your beveled pieces would be 2 x 40 x Sin(11.25/2) = 7.84".

    Hope that helps.

    Brian Walter

  8. #8
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    http://www.sawmillcreek.org/showthread.php?t=111665

    this table will make you job easy without all the math........
    Dave

    IN GOD WE TRUST
    USN Retired

  9. #9
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    Quote Originally Posted by Brian Walter View Post
    Rick,
    For starters 8 pieces with an 11.25 degree bevel will only give you 1/4 of a circle, not 1/2.

    11.25 degrees is actually correct. You have to add the two cut pieces together, which gives you the 22.5 degree total.

  10. #10
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    A minor issue with using circumference is the boards are flat, not rounded.
    Don't suppose it would make much difference however.

    What you have are 8 triangles with the sides being the radius and the base being the desired "width" of the board.

    A quick google of "triangle calculator" gave me this:
    http://ostermiller.org/calc/triangle.html

    Drop 78.75 into the 2 base angles (90 - 11.25), and the desired radius into the 2 sides, the top angle will calc out to 22.5 (correct) and give the desired base "length"

    the radius dimension should be to the outside of the circle, to allow for cutting the 11.25 angle.

    (this is assuming the desired outside radius is to the "points" not the flats)
    Last edited by Doug Arndt; 06-18-2009 at 1:49 PM.

  11. #11
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    Unless I miss my guess, both Brians are wrong. First Brian is dividing half the circumference by 8. That would be the length along the arc, not along a straight line width. This is a chordal distance which is in relation to circumference/2/8 but not the same.

    Second Brian is missing the fact that each bevel is 11.25 degrees, so when 2 are put together they equal 22.5 degrees. 8*22.5+180.

    Matt and Brian were close but the right formula is really
    length= 2(sin11.25)*rad. Matt was wrong when he said Brian's answer was easier because wrong is not easier. Other Brian was right when he said Grant was wrong.

    Brian Walter's last formula was right up until he divided the sin11.25 by 2.The correct answer is 15.607"

    After all this I hope I am right!

  12. #12
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    Quote Originally Posted by Robert McGowen View Post
    11.25 degrees is actually correct. You have to add the two cut pieces together, which gives you the 22.5 degree total.
    I'm guessing we aren't talking about the same thing when we talk bevel. I'm referring to measuring the angle between opposite sides or ends of a board. If we are talking a board with an 11.25 degree bevel on each end/side which I would call a 22.5 degree bevel then you are correct, 8 boards will give you the 1/2 circle. If that is the case, then you simply wouldn't divide the angle in half in the formula that I gave.

    Brian Walter

  13. #13
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    Here's the formula. Personally, I use CAD so I don't have to think.

    -Jeff

    Thank goodness for SMC and wood dough.

  14. #14
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    Jeff, your picture certainly makes it easier for everyone to be talking about the same thing. Greg is correct if he is measuring the bevel as Jeff shows it. I was measuring the central angle from each end of "W", not 1/2 of "W". Now wasn't that fun?

    Brian Walter

  15. #15
    I just drew it in SketchUp and measured it there.

    This is based on a 10" radius:

    Untitled.jpg
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