On the previous page, we stated that the perimeter of the rectangular gap is constant, and went on to state that for a rectangle with constant perimeter, the largest area occurs when it is a square.

On this page we examine why the square has the largest area.

In effect, we want to show that if one side of a square decreases by a given amount and the other side increases by the same amount, its area decreases. Consider the two rows in the diagram, above.

In the first row, reading from left to right, the square 'leg' decreases, so the 'gap' gets wider but shorter - its area gains by the green region but decreases by the slightly larger red region. [Note: we don't really need to include the 'leg' in these diagrams - however, by doing so we don't need to make explicit use of the as-yet unproved claim that the perimeter of the gap remains constant.]

In the second row, reading from left to right, the square 'leg' increases, so the 'gap' gets taller but narrower - its area gains by the green region but decreases by the slightly larger red region.

Put algebraically, when we change a w by w square into a w+a by w–a rectangle, its perimeter stays the same but its area decreases: (w + a)(w – a) = w^2 – a^2 < w^2.

NOTE: We first developed the basic leg-and-adjacent-gap task on the ICCAMS project, when we were looking for 'dynamic' ways of using diagrams for school algebra, in contrast to the static way they tend to be used in textbooks. The original ICCAMS task can be found here.

NOTE 2: A nice challenge is to extend the 'permitted' values of w, eg to allow negative values - what happens to the various letter-shapes and to their areas?

the end