Hexagon Scaling

Imagine the hexagon inside of a box. Say that 50% of the area of the box is the hexagon.

Now, we can clearly see that making the box 5 times larger will increase the total area by 25 times.

Since the hexagon in the box is still 50% of it's area, the hexagon also must have gotten 25 times larger. 3*25=75.

(This also applies to any 2D shape. Scaling a 3D shape based off a line, you would increase it's size by 125, and so on.)

Let $a$ be the area of the original hexagon, $x$ be the side length of the original hexagon and $A$ be the area of the larger hexagon. Note that the areas of similar plane figures have the same ratio as the squares of any two corresponding sides.

By similar figures, we have

$\dfrac{a}{A}=\dfrac{x^2}{(5x)^2}$ $\implies$ $\dfrac{3}{A}=\dfrac{1}{25}$ $\implies$ $A=3(25)=$ $\boxed{75}$

If the dimensions of an object are increased $5$ times, its area is increased $5\times5=25$ times. The shape of the object is irrelevant.