To determine the surface area ratio of similar shapes, you need to square the ratio of their corresponding linear dimensions.

When working with similar shapes, the corresponding linear dimensions—such as lengths, widths, or heights—are proportional. This means that if you have the ratio of any pair of corresponding linear dimensions, you can derive the ratio of their surface areas by squaring the linear ratio. For instance, if the ratio of the lengths of two similar shapes is $2:3$, then the ratio of their surface areas is given by

To illustrate this concept further, consider two similar cubes. Let the edge length of the smaller cube be $2 \, \text{cm}$ and the edge length of the larger cube be $4 \, \text{cm}$. The ratio of their edge lengths can be expressed as

which simplifies to

To calculate the surface area ratio, we square this linear ratio:

Thus, the surface area of the larger cube is four times greater than that of the smaller cube.

This method is applicable to any pair of similar shapes, whether they are triangles, rectangles, spheres, or any other geometric figures. The key steps are to identify the corresponding linear dimensions, calculate their ratio, and then square that ratio to obtain the surface area ratio. This principle stems from the fact that area scales with the square of the linear dimensions in similar shapes.