Question : The ratio of the area of a regular hexagon and an equilateral triangle having the same perimeter is:
Option 1: $2:3$
Option 2: $6:1$
Option 3: $3:2$
Option 4: $1:6$
Correct Answer: $3:2$
Solution :
Given: A regular hexagon and an equilateral triangle having the same perimeter.
Let the perimeter of the triangle and the hexagon be $6a$ units.
So, each side of the hexagon = $a$ units
And each side of the triangle = $2a$ units
We know that the area of a regular hexagon = $\frac{3\sqrt3}{2}a^2$ and the area of an equilateral triangle = $\frac{\sqrt3}{4}b^2$, where $a$ and $b$ are the sides of the hexagon and the equilateral triangle, respectively.
So, the area of the hexagon = $\frac{3\sqrt3}{2}a^2$ sq. units
And the area of the equilateral triangle = $\frac{\sqrt3}{4}(2a)^2$ sq. units
$\therefore$ The ratio of their area $=\frac{3\sqrt3}{2}a^2 : \frac{\sqrt3}{4}(2a)^2 = \frac{3}{2}:1 = 3:2$
Hence, the correct answer is $3:2$.
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