Question

The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.

Answer 1

Step 1. Find the radius of the moon and earth

Let the diameter of earth $=dkm$,

so the radius $\left(r_1\right)=\frac{d}{2}km$

Therefore diameter of moon $=\frac{d}{4}km$ and so radius of moon $\left(r_2\right)=\frac{d}{8}km$

Step 2. Find the surface area of the moon and earth

Assuming the earth and the moon in the shape of perfect sphere.

surface area of earth $=4{{\mathrm{\pi r}}_1}^2$ [As the area of sphere $=4{\mathrm{\pi r}}^2$]

$$\begin{gathered} =4\mathrm{\pi }{\left(\frac{\mathrm{d}}{2}\right)}^2 \\ =4\mathrm{\pi x}\frac{{\mathrm{d}}^2}{4} \\ ={\mathrm{\pi d}}^2..........................\left(i\right) \end{gathered}$$

and surface area of moon $=4{{\mathrm{\pi r}}_2}^2$

$$\begin{gathered} =4\mathrm{\pi }{\left(\frac{\mathrm{d}}{8}\right)}^2 \\ =4\mathrm{\pi x}\frac{{\mathrm{d}}^2}{64} \\ =\frac{{\mathrm{\pi d}}^2}{16}..........................\left(ii\right) \end{gathered}$$

Step 3. Find the ratio between surface area

Ratio between their surface area $={\mathrm{\pi d}}^2:\frac{{\mathrm{\pi d}}^2}{16}$

$=1:16[\mathrm{multiplying}\mathrm{by}\frac{16}{{\mathrm{\pi d}}^2}]$

Hence the required ratio is $1:16$.