The square

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The area of the square is: $A = l^{2}$

Its perimeter is: $P = 4 \cdot l$

And, applying the Pythagorean theorem, its diagonal is: $d = \sqrt{2 \cdot l^{2}} = l \cdot \sqrt{2}$

Example

Calculate the area of a square which diagonal is $3$ cm.

$3^{2} = l^{2} + l^{2} = 2 \cdot l^{2}$ $l^{2} = \frac{9}{2}$ $l = \frac{3}{\sqrt{2}} c m$

$A = (\frac{3}{\sqrt{2}})^{2} = \frac{9}{2} c m^{2}$