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 \(2\sqrt{6}\)
