*Regular hexagon + multiples of 4, a≠b1
6+4n→6→10→14→18→22→26→30→34→……Infinite polygon≠ circle.
For regular polygons that are multiples of 6+4, the height bd = diameter R is a fixed value.
360/dn (number of sides of the polygon) → sin(360/dn × ½) × 2 = side a
→ cos(360/dn × ½) × 2 = ac
Therefore, the area ratio is base a = cos(360/dn × ½) × 2 and height b = R
Area ratio of a regular hexagon (6, 12-sided polygon = 0.75)
Area of a regular hexagon=(sin30×2)²/4×tan60×6=
2.5980762113533159402911695122588
d6/ab=d6/a(cos30×2)×b(2)=0.75
Area of a regular decagon=(sin18×2)²/4×tan72×10=
2.9389262614623656458435297731954
An area ratio is a polygon that exists within a square, and a regular polygon is a shape that exists within a circle.
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Odd Regular Polygons a²=b, √b=a, a (base) b (height)
Calculating the area of a regular polygon is simple. However, for polygons, the amount of calculation increases as the number of sides increases. This becomes easier when calculated using the area ratio. Furthermore, for odd regular polygons, instead of calculating both the height b and the length of the base a, you can simply calculate the height b, which is much simpler than the base a, and then apply the odd polygon rule a²=b and √b=a to the base b to find the area ratio very simply.
The conditions a²=b and √b=a are formed only when a < 1 and b < 1.
Regular pentagon height b = 360/5 = 72 → r(0.5) + h(0.5 × cos 36)
Regular heptagon height b = 360/7 = 51.4285 → r(0.5) + h(0.5 × cos (360/7½))
Regular nonagon height b = 360/9 = 40 → r(0.5) + h(0.5 × cos 20)
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