Question

What is the Side length

Answer 1

The formula of a diameter d of a square with side a:

$d=a\sqrt2$

Substitute $d=\sqrt6\ mm$:

$a\sqrt2=\sqrt6$        multiply both sides by $\sqrt2$

$2a=\sqrt{12}$         used $\sqrt{a}\cdot\sqrt{a}=a$ and $\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}$

$2a=\sqrt{4\cdot3}$      use $\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}$

$2a=\sqrt4\cdot\sqrt3$

$2a=2\sqrt3$     divide both sides by 2

$a=\sqrt3\ mm$

The perimeter: $P=4a$

Substitute:

$P=4\sqrt3\ mm$

Answer: D.

Other method.

Use the Pythagorean theorem:

$a^2+a^2=(\sqrt6)^2$     use $(\sqrt{a})^2=a$

$2a^2=6$           divide both sides by 2

$a^2=3\to a=\sqrt3\ mm$

$P=4a\to P=4\sqrt3\ mm$