When the scale factor of enlargement is n, how much larger is the area of the shape?

When a shape undergoes an enlargement by a scale factor of n, every linear dimension of the shape, such as its length and width, is multiplied by n. For example, if the original dimensions of a two-dimensional shape are length and width, after enlargement, the new dimensions would be n times the original length and n times the original width.

The area of a two-dimensional shape is calculated as length multiplied by width. Therefore, if the initial area is represented as A (Area = length × width), after enlargement, the area becomes:

Area after enlargement = (n × length) × (n × width) = n² × (length × width) = n² × A

This indicates that the new area is n² times the original area, highlighting that the area grows in proportion to the square of the scale factor. Thus, when the scale factor of enlargement is n, the area of the shape increases by a factor of n².