Video Transcript
A rectangle has width 𝑤 and length
𝑙. A new rectangle is formed which has
the same length but double the width. Find the perimeter 𝑃 and area 𝐴
of this new rectangle.
Let’s begin by considering the
rectangle with width 𝑤 and length 𝑙. We know that each pair of parallel
sides in a rectangle are equal in length. Each of the angles in a rectangle
is equal to 90 degrees. We also recall that we can
calculate the perimeter of any rectangle by adding the length and width and then
multiplying by two. Distributing the parentheses or
expanding the brackets means that this is the same as two multiplied by the length
plus two multiplied by the width, two 𝑙 plus two 𝑤. The area of any rectangle can be
calculated by multiplying its length by its width. These are standard formulas that we
need to be able to recall.
We are told that our new rectangle
has the same length but double the width. This means that the length is still
equal to 𝑙, whereas the width is equal to two 𝑤. Substituting these into our formula
for perimeter, we see that the perimeter is equal to two multiplied by 𝑙 plus two
𝑤. Distributing the parentheses here
gives us two multiplied by 𝑙 and two multiplied by two 𝑤. This is equal to two 𝑙 plus four
𝑤. As we want a formula for the
perimeter 𝑃, 𝑃 is equal to two 𝑙 plus four 𝑤.
As already mentioned, we know that
the area of any rectangle is equal to its length multiplied by its width. For this new rectangle, this is
equal to 𝑙 multiplied by two 𝑤. This can be written as two 𝑙𝑤 or
two 𝑤𝑙. We always put the number first, but
the letters or variables can go in either order. The formula for the area 𝐴 is 𝐴
is equal to two 𝑙𝑤. If we were given specific values
for 𝑙 and 𝑤, we could then substitute these in to calculate the value of 𝑃 and
𝐴.
