1. Joined
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    09 Oct '10 08:51
    I have a regular decagon, and inside it I have drawn the largest square that will fit.

    What it the ratio:

    Area_Of_Square / Area_of_Decagon
  2. ALG
    Joined
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    09 Oct '10 12:15
    0,6639
  3. ALG
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    09 Oct '10 13:27
    Originally posted by Thomaster
    0,6639
    No, that must be wrong.
  4. Joined
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    10 Oct '10 21:30
    Originally posted by Thomaster
    No, that must be wrong.
    Yes, I think it is wrong too, but quite close.
  5. ALG
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    10 Oct '10 21:36
    Originally posted by iamatiger
    Yes, I think it is wrong too, but quite close.
    That is because my square was a diamont.
  6. Joined
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    11 Oct '10 07:34
    Originally posted by Thomaster
    That is because my square was a diamont.
    Ah, yes. I got a diamond at first too.
  7. Joined
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    11 Oct '10 11:481 edit
    Are you sure? I'm probably missing something, but I got the same answer with something that I'm pretty sure is a square. (Though I'm not certain it's the largest square, admittedly).

    Presumably that 0.6639 is an approximation of (cos 9)^2/(5 sin 18 cos 18)
  8. Joined
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    11 Oct '10 13:49
    To provide more detail, my "quadrilateral" is constructed as follows:

    If the decagon has a vertex at 0 degrees (and 36, 72, etc), then the square has vertices at 9, 99, 189, 279 degrees.

    If the answer isn't right, where I went wrong must be:
    - that isn't a square
    - that isn't the largest square
    - miscalculating the areas

    Which is it?
  9. Joined
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    12 Oct '10 00:515 edits
    Originally posted by mtthw
    If the answer isn't right, where I went wrong must be:
    - that isn't a square
    - that isn't the largest square
    - miscalculating the areas

    Which is it?
    I think you miscalculated the areas, unless I'm wrong (The fourth possibility!).
  10. Joined
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    12 Oct '10 09:33
    Originally posted by iamatiger
    I think you miscalculated the areas, unless I'm wrong (The fourth possibility!).
    OK, I'll have another look at it later. Ta.
  11. Joined
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    12 Oct '10 10:531 edit
    <slaps head/>

    Silly mistake 🙂.

    0.6310 any closer?
  12. Joined
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    12 Oct '10 20:16
    Originally posted by mtthw
    <slaps head/>

    Silly mistake 🙂.

    0.6310 any closer?
    Now only the 4th decimal place is wrong I think.

    Can you give your expression for the answer?
  13. Joined
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    13 Oct '10 08:421 edit
    Originally posted by iamatiger
    Now only the 4th decimal place is wrong I think.

    Can you give your expression for the answer?
    I get:

    cos 18/(5 sin 18 cos^2 9)

    = 0.6309778... = 0.6310 to 4 d.p. (according to this calculator 🙂)
  14. ALG
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    19 Oct '10 14:571 edit
    1 / (20 × sqrt( 5 + 2sqrt(5)) × sin² 9) = 0,6639

    Same answer. 😳
  15. Joined
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    02 Dec '10 21:2324 edits
    Sorry I forgot about this

    you are both still slightly wrong:

    Reveal Hidden Content
    From the wikipedia entry on decagons:

    Reveal Hidden Content
    Area of a regular decagon = 2.5dt where:

    Reveal Hidden Content
    t = length of a side

    Reveal Hidden Content
    d = distance between two parallel sides

    Reveal Hidden Content
    they are related:

    Reveal Hidden Content
    d = 2t(cos(54) + cos(18)) {angles in degrees}

    Reveal Hidden Content
    so area_of_decagon = 5t^2(cos(54) + cos(18))


    Reveal Hidden Content
    lets say the side is length one, then:

    Reveal Hidden Content
    area 5(cos(54) + cos(18))


    Reveal Hidden Content
    Now a decagon has 10 sides, and a square has 4, so there should be 2.5 decagagon sides between any two square corners. Draw a decagon.


    Reveal Hidden Content
    Try setting a square corner to be a decagon corner,

    Reveal Hidden Content
    now drawing corners at

    Reveal Hidden Content
    2.5 side spacings we see that even though the sides are the

    Reveal Hidden Content
    same length, the diagonals of the &quot;square&quot; are different lengths


    Reveal Hidden Content
    so it isnt actually a square


    Reveal Hidden Content
    Shunt every square corner round

    Reveal Hidden Content
    by a quarter of a side, and we can see that all the diagonals

    Reveal Hidden Content
    will now be the same, and the sides are the same, so its a square.


    Reveal Hidden Content
    By pythagoras its easy to see that the length of a diagonal

    Reveal Hidden Content
    of this square is

    Reveal Hidden Content
    sqrt(d^2 + (t/2)^2)


    Reveal Hidden Content
    where d and t are as above


    Reveal Hidden Content
    the area of a square, in terms of its diagonal length is:

    Reveal Hidden Content
    diagonal_length^2/2


    Reveal Hidden Content
    so the area of the square is:

    Reveal Hidden Content
    d^2/2 + t^2/8


    Reveal Hidden Content
    recalling that d = 2t(cos(54) + cos(18)) {angles in degrees} and that t = 1


    Reveal Hidden Content
    area_of_square = 2(cos(54) + cos(18))^2 + 1/8


    Reveal Hidden Content
    so the ratio:

    Reveal Hidden Content
    square_area/decagon_area


    Reveal Hidden Content
    (2(cos(54) + cos(18))^2 + 1/8)/5(cos(54) + cos(18))


    so the answer is:
    Reveal Hidden Content
    0.6317826922466960
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