odd regular polygon
b=r+h(0.5×cos(A½)
√b=a
Regular pentagon height √b = 0.95105651629515357211643933337938
Regular heptagon height √b = 0.97492791218182360701813168299393
Regular nonagon height √b = 0.98480775301220805936674302458952
a²=b
Regular pentagon base a²=0.90450849718747371205114670859141
Regular heptagon base a²=0.95048443395120956311805115975372
Regular nonagon base a²=0.96984631039295419202705463866237
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d=0.5⨯cos54⨯2=cos54(sin36)
a=cos54⨯sin54⨯2=0.951056516
b=0.5+h(0.5⨯sin54)=0.904508497
a=(d(cos54)⨯cos36)⨯2
b=(d(cos54)⨯cos18)+(d⨯cos54
b=r+h(0.5×cos(A½)
d/ab=0.69098300562505257589770658281718
base, height, and area ratio
---Polygon area ratio
The regular polygon d is inside the circle, and the area ratio polygon d is inside the square.
Twin area ratio: hexagon 0.75, dodecagon 0.75
Solution: ab × d = area of the polygon
4n (multiples of 4)→8→12→16→20→24→28→32→… … →infinite polygon=circle.
* Regular hexagon + multiples of 4, a=cos(A½), b=1
6+4n→6→10→14→18→22→26→30→34→… … infinite polygon≠circle.
a=cos(360/dn×½)
*Odd regular polygon b=cos(A½)+0.5, √b=a, a²=b
5→7→11→13→15→17→19→21→23→… … Infinite polygon ≠ circle.
b=r+cos(360/dn×½)
5d 0.69098300562505257589770658281718
6d 0.75
7d 0.73825040554557154454990622355428
8d √0.5
9각형(9d) 0.75712161028059446289959493451188
10d 0.77254248593736856025573354295705
11d 0.76655082780756855771796447213852
12d 0.75
13d 0.7551751545711238866910249195913
18d 0.78141679950118656983272482046192
24d 0.77645713530756228704669651287214
a=cosA, b=1
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Area Ratio Test
a2:b2=2²a²
Pentagon=2²×5d(0.69098300562505257589770658281718)=
2.7639320225002103035908263312687
Hexagon=2²×6d(0.75)=3
heptagon=2²×7d(0.73825040554557154454990622355428)=
2.95300 16221822861781996248942171
8-gon = 2²×8d(√0.5)=√8
9-gon = 2²×9d(0.75712161028059446289959493451188)=
3.0284864411223778515983797380475
12-gon = 2²×12d(0.75)=3
∞-gon (circle) = 2²×∞d(0.25π)=π
compression a2:b1
a2×b1=2a²
Pentagon=2×5d(0.69098300562505257589770658281718)=
1.3819660112501051517954131656344
Hexagon=2×6d(0.75)=1.5
heptagon=2×7d(0.73825040554557154454990622355428)=
1.4765008110911430890998124471086
8-gon=2×8d(√0.5)=√2
9-gon =2×9d(0.75712161028059446289959493451188)=
1.5142432205611889257991898690238
12-gon=2²×12d(0.75)=1.5
∞-gon(ellipse)=2×∞d(0.25π)=0.5π
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