Area of a Parallelogram: Finding Area based off Perimeter and Vice Versa

First, let's remember that the perimeter of a parallelogram is the sum of its sides,

which is

AB+BC+CD+DA

We recall that in a parallelogram, opposite sides are equal, so
BC=AD=6

Let's substitute in the formula:

2AB+12=47

2AB=35

AB=17.5

Now, after finding the missing sides, we can continue to calculate the area.

Remember, the area of a parallelogram is side*height to the side.

17.5*8= 140

To solve this problem, we aim to find the perimeter given only the area and a side of the parallelogram. The key formula for a parallelogram’s area is $\text{Area} = \text{Base} \times \text{Height}$. The perimeter of the parallelogram is calculated as $2 \times (\text{Base} + \text{Side})$.

However, the problem only provides the area and one side length and lacks information about the height or the other side. This shortage of detail restricts us from precisely determining other necessary values, like the base and the height, critical for calculating the perimeter.

Without assuming or being provided additional information, such as the height of the parallelogram or the lengths of both pairs of opposite sides, the problem lacks sufficient detail for solving explicitly. Consequently, it is impossible to calculate the perimeter from the given information alone.

Therefore, the correct conclusion is that the perimeter calculation cannot proceed with the available data.

It is not possible to calculate.

To solve this problem, we will follow these steps:

• Step 1: Setup and solve the equations for side lengths $AB$ and $AD$.
• Step 2: Calculate the area using the base $AD$ and the given height of 2 cm.

Let's begin:

Step 1: Calculate side lengths

Given that the perimeter is 22 cm, we have:

The equation simplifies to:

We are also given:

Substitute this in the first equation:

Now, substitute $AD = 8$ back into the expression for $AB$:

Step 2: Calculate the area

With $AD = 8$ cm as the base (since the problem specifies height to $AD$) and the given height of 2 cm, the area is calculated as:

Therefore, the area of the parallelogram is 16 cm².

In the first step, we must find the length of EC, which we will identify with an X.

We know that the perimeter of a rectangle is the sum of all its sides (AE+EC+CF+FA),

Since in a rectangle the opposite sides are equal, the formula can also be written like this: 2AE=2EC.

We replace the known data:

$2\times8+2X=24$

$16+2X=24$

We isolate X:

$2X=8$

and divide by 2:

$X=4$

Now we can use the Pythagorean theorem to find EB.

(Pythagoras: $A^2+B^2=C^2$)

$EB^2+4^2=5^2$

$EB^2+16=25$

We isolate the variable

$EB^2=9$

We take the square root of the equation.

$EB=3$

The area of a parallelogram is the height multiplied by the side to which the height descends, that is$AB\times EC$.

$AB=\text{ AE}+EB$

$AB=8+3=11$

And therefore we will apply the area formula:

$11\times4=44$

Let's call CE as X

According to the data

$AB=x+2,AD=x-3$

The perimeter of the parallelogram:

$2(AB+AD)$

$38=2(x+2+x-3)$

$38=2(2x-1)$

$38=4x-2$

$38+2=4x$

$40=4x$

$x=10$

Now it can be argued:

$AD=10-3=7,CE=10$

The area of the parallelogram:

$CE\times AD=10\times7=70$

The perimeter of the parallelogram is calculated as follows:

$S_{ABCD}=AB+BC+CD+DA$ Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC

According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued that$AB=DC=2BC$

We inut the data we know in the formula to calculate the perimeter:

$P_{ABCD}=2BC+BC+2BC+BC$

We replace the given perimeter in the formula and add up all the BC coefficients accordingly:

$24=6BC$

We divide the two sections by 6

$24:6=6BC:6$

$BC=4$

We know that$AB=DC=2BC$We replace the data we obtained (BC=4)

$AB=DC=2\times4=8$

As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4

To find EC we use the formula:$A_{ABCD}=AB\times EC$

We replace the existing data:

$24=8\times EC$

We divide the two sections by 8$24:8=8EC:8$

$3=EC$