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

What it the ratio:

Area_Of_Square / Area_of_Decagon

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11 Oct '10 13:49To 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

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02 Dec '10 21:2324 editsSorry I forgot about this

you are both still slightly wrong:

Reveal Hidden ContentFrom the wikipedia entry on decagons:

Reveal Hidden ContentArea of a regular decagon = 2.5dt where:

Reveal Hidden Contentt = length of a side

Reveal Hidden Contentd = distance between two parallel sides

Reveal Hidden Contentthey are related:

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

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

Reveal Hidden Contentlets say the side is length one, then:

Reveal Hidden Contentarea 5(cos(54) + cos(18))

Reveal Hidden ContentNow 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 ContentTry setting a square corner to be a decagon corner,

Reveal Hidden Contentnow drawing corners at

Reveal Hidden Content2.5 side spacings we see that even though the sides are the

Reveal Hidden Contentsame length, the diagonals of the "square" are different lengths

Reveal Hidden Contentso it isnt actually a square

Reveal Hidden ContentShunt every square corner round

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

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

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

Reveal Hidden Contentof this square is

Reveal Hidden Contentsqrt(d^2 + (t/2)^2)

Reveal Hidden Contentwhere d and t are as above

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

Reveal Hidden Contentdiagonal_length^2/2

Reveal Hidden Contentso the area of the square is:

Reveal Hidden Contentd^2/2 + t^2/8

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

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

Reveal Hidden Contentso the ratio:

Reveal Hidden Contentsquare_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 Content0.6317826922466960