Verified Answer :
Correct answer is \(36\sqrt{3}\)
The radius of an equilateral triangle is equal to 2/3 of the height.
\(R= \frac{2}{3} h= \frac{2}{3} \frac{a \sqrt{3} }{2} = \frac{a \sqrt{3} }{3}
\\4 \sqrt{3}= \frac{a \sqrt{3} }{3}
\\3\times 4\sqrt{3}=a \sqrt{3}
\\a=12
\\
\\A= \frac{a^2 \sqrt{3} }{4} =\frac{12^2 \sqrt{3} }{4} =\frac{144 \sqrt{3} }{4} =36 \sqrt{3}\)
Answer: The area of the equilateral triangle is 36√3 square meters.
Step-by-step explanation: We are to find the area of an equilateral triangle having the length of the radius equal to \(4\sqrt 3~\textup{meter}.\)
We know that the area of an equilateral triangle having length of each side equal to 'a' units is given by
\(A=\dfrac{\sqrt3}{4}a^2~\textup{sq. units.}\)
Now, the radius of an equilateral triangle is equal to two-third of the height of the trinagle.
So, if 'r' is the radius and 'h' is the height of the triangle, then we have
\(r=\dfrac{2}{3}h~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
And, the height of the equilateral triangle with side length 'a' units is given by
\(h=\dfrac{\sqrt3}{2}a.\)
So, from equation (i), we have
\(r=\dfrac{2}{3}\times \dfrac{\sqrt3}{2}a\\\\\Rightarrow 4\sqrt3=\dfrac{a}{\sqrt3}\\\\\Rightarrow a=12.\)
That is, side length, a = 12 meter.
Therefore, the area of the equilateral triangle is given by
\(A\\\\\\=\dfrac{\sqrt3}{4}a^2\\\\\\=\dfrac{\sqrt 3}{4}(12)^2\\\\\\=36\sqrt3~\textup{sq. meters.}\)
Thus, the area of the equilateral triangle is 36√3 square meters.