The smallest perimeter that a rectangle with area of 4 square units can have is ______ units. (Answer in integer)

The correct answer is 8.

To find the smallest perimeter of a rectangle with a given area, we can use the properties of rectangles and basic optimization or algebraic relationships.

Let the length of the rectangle be _l and the width be _w.

We are given that the area ($A$) of the rectangle is 4 square units:
$A=l\times w=4$

The perimeter ($P$) of a rectangle is given by the formula:
$P=2(l+w)$

By the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality), for any positive numbers _l and _w:
$\frac{l+w}{2}\ge \sqrt{l\times w}$

Multiplying both sides by 2 gives:
$l+w\ge 2\sqrt{l\times w}$

Substituting the given area ($l\times w=4$) into the inequality:
$l+w\ge 2\sqrt{4}$
$l+w\ge 2\times 2$
$l+w\ge 4$

Now, we substitute this back into the perimeter formula:
$P=2(l+w)\ge 2\times 4$
$P\ge 8$

The equality (and thus the minimum perimeter) holds when the rectangle is a square, meaning _{l = w}.
Since $l\times w=4$, we have $l^2=4$, which gives _{l = 2} and _{w = 2}.
For this square, the perimeter is:
$P=2(2+2)=8$ units.

Therefore, the smallest perimeter that the rectangle can have is 8 units.