We can think of the above diagram as showing the 'leg' and the 'gap' of the 1 unit wide P-shape. We want to know the value of w when the area of the 'gap' is a maximum.

One, rather heavy-handed, approach is to use calculus to show that w = 0.5, as outlined above, left (in red).
Another approach is outlined above, right (in blue). This might seem rather obscure, but is based on the elegant diagram used to prove Proposition 5 in Book 2 of Euclid's Elements.

However, we can use a simpler, and perhaps more insightful, geometric argument. This hinges on the elegant fact that the perimeter of the rectangular gap is constant. (Can you see why?) In turn, this means that the maximum area of the rectangle occurs when it is a square (ie when w = 1 – w, ie w = 0.5). We examine this argument more closely on the NEXT PAGE.