Decagon

From the wikipedia entry on decagons:

Area of a regular decagon = 2.5dt where:

t = length of a side

d = distance between two parallel sides

they are related:

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

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

lets say the side is length one, then:

area 5(cos(54) + cos(18))

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.

Try setting a square corner to be a decagon corner,

now drawing corners at

2.5 side spacings we see that even though the sides are the

same length, the diagonals of the "square" are different lengths

so it isnt actually a square

Shunt every square corner round

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

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

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

of this square is

sqrt(d^2 + (t/2)^2)

where d and t are as above

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

diagonal_length^2/2

so the area of the square is:

d^2/2 + t^2/8

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

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

so the ratio:

square_area/decagon_area

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

0.6317826922466960