30-gon

At a regular 30-gon the radius of the inscribed circle is 15cm. Find the "a" side size, circle radius "R", circumference, and content area.

Correct result:

a =  3.1531 cm
R =  15.0826 cm
o =  94.5938 cm
S =  709.4536 cm2

Solution:

r=15 n=30 A=360/n=360/30=12 cos⁡A/2=r/R R=r/cos⁡(A/2)=15/cos⁡(12/2)≐15.0826 (a/2)2=R2−r2 x=R2−r2=15.08262−152≐1.5766 a=2⋅ x=2⋅ 1.5766=3.1531 cm
o=n⋅ a=30⋅ 3.1531=94.5938 cm
S1=a⋅ r/2=3.1531⋅ 15/2≐23.6485 S=n⋅ S1=30⋅ 23.6485=709.4536 cm2

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