Square whose Perimeter equals its Area

From ProofWiki
Jump to navigation Jump to search

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:

$A = 16 = P$

$\blacksquare$


Historical Note

This result was known to the Pythagoreans.


Sources