Pentagon

Calculate the length of side, circumference and area of a regular pentagon, which is inscribed in a circle with radius r = 6 cm.

Correct result:

a =  7.053 cm
o =  35.265 cm
S =  85.593 cm2

Solution:

r=6 a=2⋅ r⋅ sin⁡(π/5)=2⋅ 6⋅ sin⁡(3.1416/5)=7.053 cmr=6 \ \\ a=2 \cdot \ r \cdot \ \sin(\pi/5)=2 \cdot \ 6 \cdot \ \sin(3.1416/5)=7.053 \ \text{cm}
o=5⋅ a=5⋅ 7.0534=7053200=35.265 cm
h=r2−(a/2)2=62−(7.0534/2)2≐4.8543 S1=a⋅ h/2=7.0534⋅ 4.8543/2≐17.1185 S=5⋅ S1=5⋅ 17.1185=85.593 cm2h=\sqrt{ r^2-(a/2)^2 }=\sqrt{ 6^2-(7.0534/2)^2 } \doteq 4.8543 \ \\ S_{1}=a \cdot \ h/2=7.0534 \cdot \ 4.8543/2 \doteq 17.1185 \ \\ S=5 \cdot \ S_{1}=5 \cdot \ 17.1185=85.593 \ \text{cm}^2

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