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hexagon is equal to the area of an equilateral triangle whose perimeter is 36 inches. Find the length of a side of the regular hexagon.

Sagot :

Luanaidn
[tex]Perimter\ of\ equilateral\ triangle\ =36\\ a- \ side\ of\ triangle\\ 36=3a\ |:3\\ a=12\\\\ Area\ of\ equilateral\ triangle:\\ A=\frac{a^2\sqrt{3}}{4}\\ A=\frac{12^2\sqrt{3}}{4}\\ A=\frac{144\sqrt{3}}{4}=36\sqrt3 \\\\ Hegagon\ can\ be\ divided\ into\ 6\ equilateral\ small\ triangles.\\Area\ of\ one\ of\ them: A_s=\frac{A}{6}=\frac{36\sqrt3}{6}=6\sqrt{3}\\ s-side\ of\ equilateral\ =\ side\ of\ small\ triangle\\ A_s=\frac{s^2\sqrt{3}}{4}=6\sqrt{3}\ |*4 \\ s^2\sqrt3=24\sqrt3\ |\sqrt3\\ s^2=24\\ s=\sqrt{24}[/tex][tex]\sqrt{24}=\sqrt{4*6}=2\sqrt6\\\\ Side\ of\ hexagon\ equals\ 2\sqrt6\ inches[/tex]

Answer:

[tex]2\sqrt{6}[/tex]

Step-by-step explanation:

Perimeter of equilateral triangle = 36 inches

Formula of perimeter of equilateral triangle = [tex]3\times side[/tex]

⇒[tex]36=3\times side[/tex]

⇒[tex]\frac{36}{3} = side[/tex]

⇒[tex]12= side[/tex]

Thus each side of equilateral triangle is 12 inches

Formula of area of equilateral triangle = [tex]\frac{\sqrt{3}}{4} a^{2}[/tex]

Where a is the side .

So, area of the given equilateral triangle =  [tex]\frac{\sqrt{3}}{4} \times 12^{2}[/tex]

                                                                   =  [tex]36\sqrt{3}[/tex]

Since hexagon can be divided into six small equilateral triangle .

So, area of each small equilateral triangle = [tex]\frac{36\sqrt{3}}{6}[/tex]

                                                                   =  [tex]6\sqrt{3}[/tex]

So, The area of small equilateral triangle :

[tex]\frac{\sqrt{3}}{4}a^{2} =6\sqrt{3}[/tex]

Where a is the side of hexagon .

[tex]\frac{1}{4}a^{2} =6[/tex]

[tex]a^{2} =6\times 4[/tex]

[tex]a^{2} =24[/tex]

[tex]a =\sqrt{24}[/tex]

[tex]a =2\sqrt{6}[/tex]

Hence the length of a side of the regular hexagon is [tex]2\sqrt{6}[/tex]

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