# Regular n-gon

Which regular polygon have a radius of circumscribed circle r = 10 cm and the radius of inscribed circle p = 9.962 cm?

Result

n =  36

#### Solution:

$R=10 \ \text{cm} \ \\ r=9.962 \ \text{cm} \ \\ q=r/R=9.962/10=0.9962 \ \\ r_{35}=1 / \tan( \pi / 35 )=1 / \tan( 3.1416 / 35 ) \doteq 11.1109 \ \\ R_{35}=1 / \sin( \pi / 35 )=1 / \sin( 3.1416 / 35 ) \doteq 11.1558 \ \\ q_{35}=r_{35} / R_{35}=11.1109 / 11.1558 \doteq 0.996 \ \\ r_{36}=1 / \tan( \pi / 36 )=1 / \tan( 3.1416 / 36 ) \doteq 11.4301 \ \\ R_{36}=1 / \sin( \pi / 36 )=1 / \sin( 3.1416 / 36 ) \doteq 11.4737 \ \\ q_{36}=r_{36} / R_{36}=11.4301 / 11.4737 \doteq 0.9962 \ \\ r_{37}=1 / \tan( \pi / 37 )=1 / \tan( 3.1416 / 37 ) \doteq 11.7491 \ \\ R_{37}=1 / \sin( \pi / 37 )=1 / \sin( 3.1416 / 37 ) \doteq 11.7916 \ \\ q_{37}=r_{37} / R_{37}=11.7491 / 11.7916 \doteq 0.9964 \ \\ q_{36}=q=0.9962 \ \\ n=36$

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