Question

If the ratio of the corresponding side lengths of two similar polygons is 6:11 what is the ratio of their areas?

Answer 1

$\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &\stackrel{ratio~of~the}{Sides}&\stackrel{ratio~of~the}{Areas}&\stackrel{ratio~of~the}{Volumes}\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}$

$\bf -------------------------------\\\\ 6:11\qquad \cfrac{6}{11}\qquad \stackrel{sides'~ratio}{\cfrac{6}{11}}\qquad \qquad \stackrel{areas'~ratio}{\cfrac{6^2}{11^2}}\implies \cfrac{36}{121}$