Problem 2

A stepped shape is constructed from 1$\times$1 squares with the following properties: (i) There are no gaps or overlaps. (ii) There is an odd number of squares in the bottom row (eleven in the example shown). (iii) Each row, except the bottom one, has two fewer squares than the row immediately below it. (iv) Each square, except in the bottom row, touches two squares in the row immediately below it. (v) There is exactly one square in the top row.

Prove that $36A = (P + 2)^2$, where $A$ is the area of the shape and $P$ is the length of its perimeter.