The sides can be found by taking the square root of the area.

$\displaystyle (Area)^{1/2}=s$, where $\displaystyle s$ = side.

$\displaystyle (40)^{1/2}=6.3$. 

So the length of a side is 6.3 cm.

First, know that all the side lengths of a square are equal. Second, know that the sum of all 4 side lengths gives us the perimeter.  Thus, the square perimeter of 16 is written as

$\displaystyle 16=S+S+S+S$

$\displaystyle 16=4S$

where S is the side length of a square. Solve for this S

$\displaystyle S=\frac{16}{4}=4$

So the length of each side of this square is 4.

We will have two formulas to help us solve this problem, the area and perimeter of a square.

The area of a square is:

$\displaystyle A =$$\displaystyle l$ $\displaystyle \ast$$\displaystyle w$,

where $\displaystyle l =$ length of the square and $\displaystyle w=$ width of the square.

The perimeter of a square is:

$\displaystyle P =$$\displaystyle 2($$\displaystyle l$$\displaystyle +$$\displaystyle w$$\displaystyle )$

Plugging in our values, we have:

$\displaystyle 25$$\displaystyle m^2$$\displaystyle =$$\displaystyle l$$\displaystyle \ast$$\displaystyle w$

$\displaystyle 20$$\displaystyle m$$\displaystyle =$$\displaystyle 2($$\displaystyle l$$\displaystyle +$$\displaystyle w$$\displaystyle )$

 Since all sides of a square have the same value, we can replace all $\displaystyle l$ and $\displaystyle w$ with $\displaystyle s$ (side). Our equations become:

$\displaystyle 25$$\displaystyle m^2$$\displaystyle =$ $\displaystyle s^2$

$\displaystyle 20$$\displaystyle m$$\displaystyle =$ $\displaystyle 2s$

Therefore, $\displaystyle s =$$\displaystyle 5$$\displaystyle m$.

The area of any quadrilateral can be determined by multiplying the length of its base by its height.

Since we know the shape here is square, we know that all sides are of equal length. From this we can work backwards by taking the square root of the area to find the length of one side.

$\displaystyle \sqrt{36}=6$

The length of one (and each) side of this square is 6 inches.

One of the necessary conditions of a square is that all sides be of equal length. Therefore, because we are given the length of one side we know the length of all sides and that includes the length of the opposite side. Since the length of one of the sides is 4 we can conclude that all of the sides are 4, meaning the opposite side has a length of 4.

We begin by recalling the formulas for the perimeter and area of a square respectively.

$\displaystyle P=4s$

$\displaystyle A=s^2$

Using these formulas and the fact that the perimeter is half the area, we can create an equation.

$\displaystyle 4s=\frac{1}{2}(s^2)$

We can multiply both sides by 2 to eliminate the fraction.

$\displaystyle 8s=s^2$

To get one side of the equation equal to zero, we will move everything to the right side.

$\displaystyle 0=s^2-8s$

Next we can factor.

$\displaystyle 0=s(s-8)$

Setting each factor equal to zero provides two potential solutions.

$\displaystyle s=0$       or        $\displaystyle s-8=0$

                             $\displaystyle s=8$

However, since a square cannot have a side of length 0, 8 is our only answer.

$\displaystyle Area = Length \times Length$

$\displaystyle 100 = (Length)^2$

$\displaystyle Length = \sqrt{100}=10$

If diagonal $\displaystyle \overline{SU}$ of Square $\displaystyle SQUA$ is constructed, then $\displaystyle \bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse $\displaystyle SU = \sqrt{2x}$. By the 45-45-90 Theorem, the sidelength $\displaystyle SQ$ can be calculated as follows:

$\displaystyle SQ = \frac{SU}{\sqrt{2}} = \frac{\sqrt{2x}}{\sqrt{2}} = \sqrt{\frac{2x}{2}} = \sqrt{x}$.

The diameter of a circle with circumference 20 is

$\displaystyle \frac{20}{\pi } \approx \frac{20 }{3.1416} \approx 6.3662$

The diameter of a circle that circumscribes a square is equal to the length of the diagonals of the square.

If diagonal $\displaystyle \overline{SU}$ of Square $\displaystyle SQUA$ is constructed, then $\displaystyle \bigtriangleup SQU$ is a 45-45-90 triangle with hypotenuse approximately 6.3662. By the 45-45-90 Theorem, divide this by $\displaystyle \sqrt{2} \approx 1.4142$ to get the sidelength of the square:

$\displaystyle 6.3662 \div 1.4142 \approx 4.5$

The area of Square $\displaystyle SQUA$ is the square of sidelength $\displaystyle SQ$, or $\displaystyle (SQ)^{2}$.

The area of Rectangle $\displaystyle RECT$ is $\displaystyle RE \cdot EC$. Rectangle $\displaystyle RECT$ has area 90% of that of Square $\displaystyle SQUA$, which is $\displaystyle \frac{9}{10} (SQ)^{2}$;  $\displaystyle RE$ is 80% of $\displaystyle SQ$, so $\displaystyle RE = \frac{4}{5}SQ$. We can set up the following equation: 

$\displaystyle \frac{9}{10} (SQ)^{2} = RE \cdot EC$

$\displaystyle \frac{9}{10} (SQ)^{2} = \frac{4}{5} SQ \cdot EC$

$\displaystyle \frac{9}{10} \cdot SQ = \frac{4}{5} \cdot EC$

$\displaystyle \frac{10} {9} \cdot \frac{9}{10} \cdot SQ = \frac{10} {9} \cdot \frac{4}{5} \cdot EC$

$\displaystyle SQ = \frac{8} {9} \cdot EC$

As a percent, $\displaystyle \frac{8} {9}$ of $\displaystyle EC$ is $\displaystyle \frac{8} {9} \times 100 \%= 88 \frac{8} {9} \%$