Video: Writing Algebraic Expressions Involving Geometrical Formulas

A rectangle has width 𝑀 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.

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Video Transcript

A rectangle has width 𝑀 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 𝑀 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 𝑀. 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 𝑀. Substituting these into our formula for perimeter, we see that the perimeter is equal to two multiplied by 𝑙 plus two 𝑀. Distributing the parentheses here gives us two multiplied by 𝑙 and two multiplied by two 𝑀. This is equal to two 𝑙 plus four 𝑀. As we want a formula for the perimeter 𝑃, 𝑃 is equal to two 𝑙 plus four 𝑀.

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 𝑀. This can be written as two 𝑙𝑀 or two 𝑀𝑙. 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 𝑙𝑀. If we were given specific values for 𝑙 and 𝑀, we could then substitute these in to calculate the value of 𝑃 and 𝐴.

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