All the sides of a hexagon become three times the original length. find the ratio of areas of the new and old hexagons.

To determine the ratio, we need to know the formula of the area of an hexagon in terms of the length of its sides. We cannot directly conclude that the ratio would be 3, the same as that of the ratio of the lengths of the side, since it may be that the relationship of the area and length is not equal. The area of a hexagon is calculated by the expression:

A = (3√3/2) a^2

So, we let a1 be the length of the original hexagon and a2 be the length of the new hexagon.

A2/A1 = (3√3/2) a2^2 / (3√3/2) a1^2

A2/A1 = (a2 / a1) ^2 = 3^2 = 9

Therefore, the ratio of the areas of the new and old hexagon would be 9.