Ms. Price pays you $20 to mow her yard. After seeing you did such a good job, Mr. Sherman asks if you would mow his yard, too. Mr. Sherman's yard is twice as long and twice as wide as Ms. Price's yard, so he says he'll pay you twice as much.
Does Mr. Sherman's math make sense? How much should you charge him?
Let's look at how perimeter and area change as the dimensions of a shape are changed.
All sides have length in the square below.
The perimeter of the square is .
The area of the square is .
Now let's double the length of each side.
The perimeter is now .
After doubling each side length, the perimeter doubled from to . Length is a one-dimensional measurement, so we multiply by our scale factor of 2 once. The original length of each side was multiplied by , so we multiply the original perimeter by as well.
The area is now .
After doubling each side length, the area quadrupled from to . Area is a two-dimensional measurement, so we multiply by our scale factor twice. The original length of each side was multiplied by , so we have to multiply the original area by .