The formula of a diameter d of a square with side a:
[tex]d=a\sqrt2[/tex]
Substitute [tex]d=\sqrt6\ mm[/tex]:
[tex]a\sqrt2=\sqrt6[/tex] multiply both sides by [tex]\sqrt2[/tex]
[tex]2a=\sqrt{12}[/tex] used [tex]\sqrt{a}\cdot\sqrt{a}=a[/tex] and [tex]\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}[/tex]
[tex]2a=\sqrt{4\cdot3}[/tex] use [tex]\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}[/tex]
[tex]2a=\sqrt4\cdot\sqrt3[/tex]
[tex]2a=2\sqrt3[/tex] divide both sides by 2
[tex]a=\sqrt3\ mm[/tex]
The perimeter: [tex]P=4a[/tex]
Substitute:
[tex]P=4\sqrt3\ mm[/tex]
Answer: D.
Other method.
Use the Pythagorean theorem:
[tex]a^2+a^2=(\sqrt6)^2[/tex] use [tex](\sqrt{a})^2=a[/tex]
[tex]2a^2=6[/tex] divide both sides by 2
[tex]a^2=3\to a=\sqrt3\ mm[/tex]
[tex]P=4a\to P=4\sqrt3\ mm[/tex]
