Question

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 1

Answer:

$2\sqrt{6}$

Step-by-step explanation:

Perimeter of equilateral triangle = 36 inches

Formula of perimeter of equilateral triangle = $3\times side$

⇒$36=3\times side$

⇒$\frac{36}{3} = side$

⇒$12= side$

Thus each side of equilateral triangle is 12 inches

Formula of area of equilateral triangle = $\frac{\sqrt{3}}{4} a^{2}$

Where a is the side .

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

                                                                   =  $36\sqrt{3}$

Since hexagon can be divided into six small equilateral triangle .

So, area of each small equilateral triangle = $\frac{36\sqrt{3}}{6}$

                                                                   =  $6\sqrt{3}$

So, The area of small equilateral triangle :

$\frac{\sqrt{3}}{4}a^{2} =6\sqrt{3}$

Where a is the side of hexagon .

$\frac{1}{4}a^{2} =6$

$a^{2} =6\times 4$

$a^{2} =24$

$a =\sqrt{24}$

$a =2\sqrt{6}$

Hence the length of a side of the regular hexagon is $2\sqrt{6}$

Answer 2

$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}$$\sqrt{24}=\sqrt{4*6}=2\sqrt6\\\\ Side\ of\ hexagon\ equals\ 2\sqrt6\ inches$