Let's consider the two following similar rectangles.
We are given that the original logo has the dimensions of the larger rectangle and the new logo has the dimensions of the smaller rectangle. To explain the thinking of Robert and Denise, we will compare the perimeters and the areas of these logos. Let's start by recalling the formulas for the perimeter and area of a rectangle with length l and width w.
Perimeter of a Rectangle
2 l + 2 w
Area of a Rectangle
l w
Now we will calculate the perimeter and area of each logo using these formulas.
Logo
l, in.
w, in.
Perimeter, in.
Area, in^2
original
6
4
2( 6)+2( 4)=12+8=20
6 ( 4) = 24
new
3
2
2( 3)+2( 2)=6+4=10
3 ( 2) = 6
Finally, we compare the perimeters and the areas of the two logos. We will start by calculating the ratio of the perimeter of the new logo to the perimeter of the original logo.
Perimeter of the new logo/Perimeter of the original logo=10in./20in.=1/2
This ratio means that the perimeter of the new logo is 12 the perimeter of the original logo. Since Denise thinks that the new logo is 12 the original size, Denise thinks of size in terms of perimeter. Now let's calculate the ratio of the area of the new logo to the area of the original logo.
Area of the new logo/Area of the original logo=6in^2/24in^2=1/4
This ratio means that the area of the new logo is 14 the area of the original logo. Since Robert thinks that the new logo is 14 the original size, Robert thinks of size in terms of area.
