The areas of two similar octagons are \( 4 \mathrm{~m}^{2} \) and \( 9 \mathrm{~m}^{2} \). What is the scale factor of their side lengths? [?]: [ ]

To find the scale factor of the side lengths of similar octagons, you take the square root of the ratio of their areas. The areas given are \( 4 \mathrm{~m}^{2} \) and \( 9 \mathrm{~m}^{2} \). So, the ratio of the areas is \( \frac{4}{9} \). The scale factor for the side lengths is the square root of this ratio, which is \( \sqrt{\frac{4}{9}} = \frac{2}{3} \). Hence, the scale factor of their side lengths is \( \frac{2}{3} \). The concept of similarity in geometry can be a lot of fun! Think of it as comparing two identical, but differently sized, cheese pizzas. If one pizza is twice the size of the other, the slices (side lengths) may be smaller in one compared to the larger one, but they are still in the same proportion to each other. The knowledge of proportions helps us understand scaling in art, architecture, and even model building!