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.

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]

[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]