30-gon

The radius of the inscribed circle is 15cm at a regular 30-gon. Find the side length a, circle radius R, circumference, and area.

Correct answer:

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

Step-by-step explanation:

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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