Originally posted by GinoJ I do not think I could ever have as much as yours. 😉
[edit] So, I leave the "posers and puzzles" section to your dominance since I am just an ignorant. ðŸ˜
Have fun!
If you leave, Gino, much of the recent fun goes with you.
This fourm is very dependant of people who invent or copy problems so the rest of us put the little grey ones in ction.
One problem is better than no problem, even if a good problem is better than an ordinary one. And favorites is always welcome if they're good.
The way of solving the heptagon problem could be very like that of the hexagon. Give it a try and see where it leads.
A heptagon has 7 sides, does it not? I daresay the answer to your query will end up a great deal more complicated than the previous question, because it will involve trigonometry, and the sines and cosines of angles which have no exact answer, only approximations.
With a hexagon, you were dealing with sin(60d) which is very well known as an exact number (sqrt(3)/2)
The area of a heptagon would be 3.5 * sin(64+2/7d).
Since a regular hexagon is composed of 6 equalateral triangles you can just solve for the area of one triangle and multiply by 6:
A=Sqr root of (S*(S-a)*(S-b)*(S-c)
a, b and c all = exactly 1.
S= (a+b+c)/2
S= 3/2
S= 1.5
So, Sqr root of (1.5)*(0.5)*(0.5)*(0.5) = Sqr root (1.5)*(0.125)
= Sqr root of (0.1875) or 0.433012702 = area of equalateral triangle of side 1.
0.433012702 * 6= 2.598076211 Square units= area of regular hexagon with unit sides. So 2.59 and change is the answer.
I deliberately did it a differant way from Fabian and got exactly the same answer so what is wrong with YOUR logic?