Decagon

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

0,6639

Originally posted by Thomaster
0,6639

No, that must be wrong.

Originally posted by Thomaster
No, that must be wrong.

Yes, I think it is wrong too, but quite close.

Originally posted by iamatiger
Yes, I think it is wrong too, but quite close.

That is because my square was a diamont.

Originally posted by Thomaster
That is because my square was a diamont.

Ah, yes. I got a diamond at first too.

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)

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?

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!).

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.

<slaps head/>

Silly mistake 🙂.

0.6310 any closer?

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?

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 🙂)

1 / (20 × sqrt( 5 + 2sqrt(5)) × sin² 9) = 0,6639

Same answer. 😳