Question

Two shapes are similar and their sides have a ratio of 2:9. What is the ratio of the surface areas for

these two shapes?

Answer 1

`\bf ~\hspace{5em} \textit{ratio relations of two similar shapes} \\\\ \begin{array}{ccccllll} &\stackrel{\stackrel{ratio}{of~the}}{Sides}&\stackrel{\stackrel{ratio}{of~the}}{Areas}&\stackrel{\stackrel{ratio}{of~the}}{Volumes}\\ \cline{2-4}&\\ \cfrac{\stackrel{similar}{shape}}{\stackrel{similar}{shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array}~\hspace{6em} \cfrac{s}{s}=\cfrac{√(Area)}{√(Area)}=\cfrac{\sqrt[3]{Volume}}{\sqrt[3]{Volume}} \\\\[-0.35em] ~\dotfill`

`\bf s:s\implies 2:9\implies \cfrac{s}{s}\implies \cfrac{2}{9}=\cfrac{√(Area)}{√(Area)}\implies \cfrac{2}{9}=\sqrt{\cfrac{Area}{Area}} \\\\\\ \left( \cfrac{2}{9} \right)^2=\cfrac{Area}{Area}\implies \cfrac{2^2}{9^2}=\cfrac{Area}{Area}\implies \cfrac{4}{81}=\cfrac{Area}{Area}`