Square whose Perimeter equals its Area
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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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16$