Let's begin with recalling that a scale model is an object with lengths proportional to the object it represents. This indicates that the model object must have the same shape as the original object.
In the case of extending the side lengths by a constant, not scaling, this means that the original object must be a regular polygon.
Extra
What if the original object is not a regular polygon?
To visualize why the original object must be a regular polygon, let's see what will happen if we extend the side lengths of a right triangle, that is not a regular polygon, by 1. We will use one of the Pythagorean Triples, 3,4, and 5.
As we can see, when we extend the side lengths by 1 the model object is not a right triangle anymore. Therefore, the original object and the model object do not have the same shape.
