Two regular hexagons have sides in the ratio 3:5. The area of the smaller hexagon is 81 square metres. What is the area of the larger hexagon? Choose from 240, 125, 135 or 225 square metres.

Step-by-step solution

1. 1

   Note that all regular hexagons are similar. Having six equal sides and six equal angles pins the shape completely, so the two hexagons differ only by a scale factor. That is what licenses the similarity rules; you do not need the hexagon area formula $\tfrac{3\sqrt3}{2}s^2$ at all.

2. 2

   Recall how area behaves under scaling. If lengths scale by $k$, area scales by $k^2$, because area is a product of two lengths. Here

   $k=\frac{\text{larger side}}{\text{smaller side}}=\frac{5}{3}\quad\Longrightarrow\quad \frac{A_{\text{large}}}{A_{\text{small}}}=k^{2}=\frac{25}{9}.$

3. 3

   Multiply the known area by the squared ratio.

   $A_{\text{large}}=81\cdot\frac{25}{9}=9\cdot 25=225\ \text{m}^2.$

   The $9$ divides into $81$ exactly, which is a good sign the problem was designed for this route.

4. 4

   Confirm with the explicit formula. For a regular hexagon of side $s$, $A=\tfrac{3\sqrt3}{2}s^2$. From $81=\tfrac{3\sqrt3}{2}s^2$ we get $s^2=\tfrac{54}{3\sqrt3}=\tfrac{18}{\sqrt3}$, and the larger side satisfies $S=\tfrac53 s$, so $S^2=\tfrac{25}{9}s^2$ and $A_{\text{large}}=\tfrac{25}{9}\cdot 81=225$. Same answer, more work.

5. 5

   Eliminate the distractors. $135=81\cdot\tfrac53$ comes from scaling by the ratio instead of its square — the standard trap. $125=5^3$ and $240$ correspond to nothing in the problem. The answer is $225\ \text{m}^2$.