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

The perimeter of the rectangle equals the sum of all its sides, which means:

$P=AB+BC+CD+DA$

Since in a rectangle each pair of opposite sides are equal, we can say that:

$BC=AD=5$

This means that the two sides together equal 10, and now we'll subtract them from the perimeter and get:

$AB+DC=30-10=20$

This means sides AB and DC together equal 20, and since they are equal to each other, we'll divide 20 to find out how much each one equals:

$20:2=10$

Now we'll multiply side AB by side BC to find the area of the rectangle:

$10\times5=50$

The perimeter of the rectangle equals:

$P=AB+BC+CD+DA$

Since we know that BC equals 5 and in a rectangle opposite sides are equal to each other, we get:

$40=AB+5+CD+5$

$40=10+AB+CD$

Since AB equals CD we can write the equation as follows:

$40=2AB+10$

Let's move 10 to the other side and change the sign accordingly:

$40-10=2AB$

$30=2AB$

Let's divide both sides by 2:

$15=AB$

Now we know the length and width of the rectangle and can calculate its area:

$15\times5=75$

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:

$11\times4=44$

After squaring all sides, we can calculate the area as follows:

$4^2=16$

Since we are given that the area of the square equals the area of the rectangle , we will write an equation with an unknown since we are only given one side length of the parallelogram:

$16=1\times x$

$x=16$

In other words, we now know that the length and width of the rectangle are 16 and 1, and we can calculate the perimeter of the rectangle as follows:

$1+16+1+16=32+2=34$