# Algebraic Reasoning with Area and Perimeter

This video goes into detail explaining algebraic reasoning with area and perimeter using several examples. The examples include doubling a rectangle’s width and length, as well as doubling its length and tripling its width.

## Algebraic Reasoning with Area and Perimeter

If the length and width of a rectangle are both doubled, what happens to the rectangle’s area and its perimeter? If the length is doubled while the width is tripled, what happens to the area and to the perimeter?

We’ll start with the first question first. We’re talking about a rectangle, and area, and perimeter, so let’s start by writing down the formula for area of a rectangle and perimeter of a rectangle. The area of a rectangle is found by multiplying the length times the width, while the perimeter of a rectangle is found by multiplying 2 times the quantity of the length plus the width.

If the length and the width of a rectangle are both doubled, then that means whatever the length was, it’s now twice that. Our area would become twice the length, times twice the width, and 2 times 2 is 4, so our area would be 4 times the length, times the width, (or quadrupled).

The area would quadruple if the length and width were both doubled. Now for the perimeter, so it’s the same thing, the length and the width are going to be doubled. Our perimeter then is 2 times, 2 times the length, plus 2 times the width.

We have a GCF of 2, which we can factor out, so 2 times 2 is 4, times the quantity of the length, plus the width. That means then that the perimeter has been doubled, it’s 2 times the original. It was originally 2 times the length, plus the width, and now it’s 4 times the length, plus the width.

Now let’s look at the second question. If the length is doubled (so length is still doubled), but this time the width is tripled, what happens to the area and to the perimeter? The area– the length is doubled (so length is 2L), and the width is tripled (width is 3w). 2 length times 3 width (2 times 3 is 6), so 6 times length, times width.

If the length was doubled and the width was tripled, then the area would be 6 times greater than it was originally. Now for perimeter. The perimeter is twice (or 2 times) the length is doubled (so 2L), plus the width is tripled (3W). This time there is not a GCF for our 2 terms inside the parenthesis, so we can’t simplify this any further.

Which means really there is no relationship for perimeter (with the old figure and the new) if the length is doubled and the width is tripled.