The figure below shows a rectangle whose length is seven centimetres and width is 𝑎 centimetres. If its perimeter is 𝑝 centimetres, write a formula for 𝑝 in terms of 𝑎 and find the perimeter when 𝑎 is equal to four centimetres.
The perimeter of any shape is calculated by adding the distances around the outside of the shape. In this case, 𝑝 is equal to seven plus 𝑎 plus seven plus 𝑎. Grouping the like terms gives us two 𝑎 plus 14. As 𝑎 plus 𝑎 is equal to two 𝑎 and seven plus seven is equal to 14, therefore the formula for 𝑝 in terms of 𝑎 is 𝑝 equals two 𝑎 plus 14.
In order to find the perimeter when 𝑎 is four centimetres, we can substitute four into the equation: 𝑝 equals two multiplied by four plus 14. As two multiplied by four is equal to eight, the perimeter will be eight plus 14. Eight plus 14 is equal to 22. Therefore, the perimeter when 𝑎 equals four is 22 centimetres.
We could also have answered the second part of the question by replacing 𝑎 with four on the diagram. We could then have added seven, four, seven, and another four. This would also have given us 22 centimetres for the perimeter.
In relation to our formula for the perimeter 𝑝 equals two 𝑎 plus 14, two is a common factor. Therefore, we could factorize out the two, giving us an alternative formula for 𝑝 of two multiplied by 𝑎 plus seven as two multiplied by 𝑎 is two 𝑎 and two multiplied by seven is 14.