The ratio of corresponding sides of two regular polygons is 3:4. The area of the larger polygon is 320 square metres. What is the area of the smaller polygon? Choose from 240, 569, 427 or 180 square metres.

Step-by-step solution

1. 1

   Confirm the two figures are similar. Regular polygons with the same number of sides are always similar — equal angles, proportional sides — so the scaling rules apply without needing to know how many sides there are.

2. 2

   Decide which direction you are scaling. You are given the larger area ($320$) and asked for the smaller one, so the length factor is $\tfrac34$, less than $1$, and the answer must come out below $320$. Two of the four options ($569$ and $427$) are larger than $320$ and can be discarded on sight.

3. 3

   Square the ratio to get the area factor. Area is two-dimensional:

   $\frac{A_{\text{small}}}{A_{\text{large}}}=\left(\frac{3}{4}\right)^{2}=\frac{9}{16}.$

4. 4

   Multiply.

   $A_{\text{small}}=320\cdot\frac{9}{16}=20\cdot 9=180\ \text{m}^2.$

5. 5

   Check the remaining distractor. $240=320\cdot\tfrac34$ is what you get by scaling with the plain side ratio instead of its square — the single most common error on this question type. The correct answer is $180\ \text{m}^2$.