Square whose Perimeter equals its Area

Theorem
The $4 \times 4$ square is the only square whose area in square units equals its perimeter in units.

The area and perimeter of this square are $16$.

Proof
Let $S$ be a square whose area equals its perimeter.

Let $A$ be the area of $S$.

Let $P$ be the perimeter of $S$.

Let $b$ be the length of one side of $S$.

From Area of Square:
 * $A = b^2$

From Perimeter of Rectangle:
 * $P = 2 b + 2 b = 4 b$

Setting $A = P$
 * $b^2 = 4 b$

and so:
 * $b = 4$

and so:
 * $A = 16 = P$