Hexagon ABCDEF

In the regular hexagon ABCDEF, the diagonal AE has a length 8cm. Calculate the circumference and the hexagon area.

Correct result:

o =  27.713 cm
S =  55.426 cm2

Solution:

u=8 cm n=6 A=360/n=360/6=60 ∘  sin⁡A=u/2:a  a=u2/sin⁡A∘=u2/sin⁡60∘ =82/sin⁡60∘ =82/0.866025=4.6188  o=n⋅ a=6⋅ 4.6188=16 3=27.713 cmu=8 \ \text{cm} \ \\ n=6 \ \\ A=360/n=360/6=60 \ ^\circ \ \\ \ \\ \sin A=u/2 : a \ \\ \ \\ a=\dfrac{ u }{ 2 } / \sin A ^\circ =\dfrac{ u }{ 2 } / \sin 60^\circ \ =\dfrac{ 8 }{ 2 } / \sin 60^\circ \ =\dfrac{ 8 }{ 2 } / 0.866025=4.6188 \ \\ \ \\ o=n \cdot \ a=6 \cdot \ 4.6188=16 \ \sqrt{ 3 }=27.713 \ \text{cm}
S1=a2⋅ 34=4.61882⋅ 34≐9.2376 cm2  S=n⋅ S1=6⋅ 9.2376=32 3=55.426 cm2S_{1}=a^2 \cdot \ \dfrac{ \sqrt{ 3 } }{ 4 }=4.6188^2 \cdot \ \dfrac{ \sqrt{ 3 } }{ 4 } \doteq 9.2376 \ \text{cm}^2 \ \\ \ \\ S=n \cdot \ S_{1}=6 \cdot \ 9.2376=32 \ \sqrt{ 3 }=55.426 \ \text{cm}^2

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