# Decagon

iamatiger
Posers and Puzzles 09 Oct '10 08:51
1. 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. 09 Oct '10 12:15
0,6639
3. 09 Oct '10 13:27
Originally posted by Thomaster
0,6639
No, that must be wrong.
4. 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. 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. 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. 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. 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. 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. 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. 12 Oct '10 10:531 edit

Silly mistake ðŸ™‚.

0.6310 any closer?
12. 12 Oct '10 20:16
Originally posted by mtthw

Silly mistake ðŸ™‚.

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

13. 13 Oct '10 08:421 edit
Originally posted by iamatiger
Now only the 4th decimal place is wrong I think.

I get:

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

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

15. 02 Dec '10 21:2324 edits

you are both still slightly wrong:

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 &quot;square&quot; 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))