Question

An equilateral triangle and a regular hexagon have the same perimeter. If the area of the hexagon is $72\sqrt{3}$, calculate the area of the triangle.

• A. $62.354$
• B. $83.138$
• C. $101.823$
• D. $103.923$
• E. $124.708$

Answer 1

Answer: (B)

$83.138$

Let the length of side of equilateral triangle be $a$ and the length of side of hexagon be $b$.

Given, $3a=6b$ , which implies $a=2b$
Area of hexagon is $6\times \frac{1}{2}\times b\times b\times \sin \left(60\right)=\frac{3\sqrt{3}}{2}{b}^2=72\sqrt{3}$ , which implies
$b^2=48$

$\Rightarrow b=4\sqrt{3}$
therefore $a=2b=8\sqrt{3}$
Area of equilateral triangle is $\frac{1}{2}\times a\times a\times \sin 60$

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

$=\frac{\sqrt{3}}{4}\times 64\times 3$

$=48\sqrt{3}$

$=83.138$