Perimeter as an Expression

Sometimes a side is not a number but an expression. A rectangle is w wide, and its length is 5 more than its width, so w + 5. Its perimeter is two widths plus two lengths: 2w + 2(w + 5).

This simplifies with the tools from the earlier chapters: distribute, then collect like terms. 2(w + 5) is 2w + 10, so the perimeter is 2w + 2w + 10, and the two w terms make 4w + 10.

The short form works for every rectangle of this shape. A width of 3 gives 4 × 3 + 10, which is 22. Why simplify? The short form takes one multiplication and one addition, and it shows at a glance how the perimeter grows: by 4 for every extra unit of width.