Question

If the surface areas of two similar figured are 25cm^2 and 36cm^2, and the volume of the larger figure is 216cm^3, what is the volume of the smaller figure?

Answer 1

If two figures are similar, then there is a relation between their areas and volumes:

$\begin{gathered} \frac{A_1}{A_2}=k^2...(1) \\ \\ \frac{V_1}{V_2}=k^3...(2) \end{gathered}$

Where k is a constant. From the problem, we identify:

$\begin{gathered} A_1=25\text{ cm}^2 \\ A_2=36\text{ cm}^2 \\ V_2=216\text{ cm}^3 \end{gathered}$

Using equation (1):

$\begin{gathered} \frac{25}{36}=k^2 \\ \\ \Rightarrow k=\frac{5}{6} \end{gathered}$

Finally, using (2) to find the volume of the smaller figure:

$\begin{gathered} \frac{V_1}{216}=\frac{5^3}{6^3} \\ \\ \frac{V_1}{216}=\frac{125}{216} \\ \\ \therefore V_1=125\text{ cm}^2 \end{gathered}$