The area of a parallelogram: what is it and how is it calculated?

The parallelogram is a four-sided polygon (quadrilateral), whose opposite sides are.

Property of parallelograms

• The opposite angles of the parallelogram have the same size.
• The opposite sides of the parallelogram have the same length.
• Parallelograms have two intersecting diagonals that create two pairs of triangles. In addition, the four triangles that are formed have the same area.
• The angles of the parallelogram complement each other until they reach $180^o$ degrees.
• The sum of the squares of its diagonals is equal to the sum of the squares of the four sides of the parallelogram.

In other words:

$KM^{2}+LN^{2}=KL^{2}+LM^{2}+MN^{2}+NK^{2}$

Or, in other words:

$KM^{2}+LN^{2}=2KL^{2}+2LM^{2}$

• Rectangle: is a parallelogram in which all its angles are right angles, that is, they measure $90^o$ degrees and its two diagonals have the same length.

Rectangle

• Rhombus: a parallelogram whose four sides are of equal length (and its two diagonals intersect at right angles, that is, they are perpendicular).

Rhombus

• Square: is a parallelogram that meets the definition of rectangle and rhombus (but also its two diagonals are perpendicular and have the same length).

Square

Find the area of the parallelogram $KLMN$ illustrated in the figure below using the data provided:

• $MN=10cm$
• $KP=5cm$

Area of a Parallelogram

Exercise 1

Solution:

This is a fairly simple exercise in which we must substitute the given data in the formula corresponding to the area of a parallelogram:

$A=MN\cdot KP=10\cdot5=50cm²$

Answer: The area of the parallelogram $KLMN$ is $50cm²$.

Analyze the illustration below and indicate if there are any errors in the data given. Explain your answer.

Solution:

This exercise deals with the area of a parallelogram. As we have already said, the area of this geometric shape can be calculated in two ways. With the first one, we must use as base the side $DC$ and consider as its relative height $AS$; the other way, is to consider the adjacent side $BC$ as the base and its relative height $AF$. The answer we obtain by applying both methods must be the same.

We substitute the data in the formula and we obtain the following:

$A=DC\cdot AS=9\cdot3=27$

$A=BC\cdot AF=6\cdot5=30$

As we can see, we have obtained a different result by applying one or the other method and, therefore, the given data are wrong.

Find the area of the parallelogram $DEFG$ according to the illustration and the data below:

• $DE=12\operatorname{cm}$
• $KG=5\operatorname{cm}$
• $DK=9\operatorname{cm}$

Solution:

If we look at the illustration, we see that $DK$ refers to the external height of the parallelogram $DEFG$.

According to the characteristics of the parallelogram that we have just learned, the opposite sides of a parallelogram are identical and parallel to each other, that is: $DE= GF=12$ and $DE$ parallel to $GF$.

To calculate the area of this parallelogram we do not need the data about the length of $KG$ since this information is not useful for such a calculation, but was given to us only to confuse us. To calculate the area of a parallelogram, we only need the length of a side and its relative height.

That said, we substitute the data into the formula and we will get the following:

$A=GF\cdot DK=12\cdot9=108cm²$

Answer: The area of the parallelogram $DEFG$ is $108 cm²$.

Inside the parallelogram $ABCD$ is the rectangle $AECF$ with a perimeter of $24$.

$AE = 8$

Task:

What is the area of the parallelogram?

Solution:

In the first step we must find the length $EC$, which we will identify as $X$.

We know that the perimeter of the rectangle is equal to the sum of its sides $(AE+EC+CF+FA)$.

Because in the rectangle the opposite sides are equal, we can write the formula like this: $2AE+2EC=24$

We substitute the known data:

$2\times8+2X=24$

$16+2X=24$

We clear the $X$

$2X=8$

And divide by $2$

$X=4$

Now, we can use the Pythagorean formula to calculate $EB$.

Pythagoras -$A^2+B^2=C^2$

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

$EB^2+16=25$

We clear $EB$

$EB^2=9$

Calculate the root

$EB=3$

The area of the parallelogram is the product of the side $AB$ by its relative height EC $AB\times EC$

$AB=\text{ AE}+EB$

On the other hand,

$AB=8+3=11$

Substitute the data into the area formula:

$11\times4=44$

Answer: $44$

Given that:

The perimeter of the parallelogram $ABCD$ is equal to $22 cm$. $DL = 3cm$

$AC = 4cm$

The height $=?$

And the side of the parallelogram

$DL = 3cm$

Task:

Calculate the area of the parallelogram. $ABCD$

Solution:

Parallel opposite sides are equal $AC=BD=4cm$

Parallel opposite sides are equal $AB=CD=Xcm$

$AB+BD+CD+AC\text{ }=$ Perimeter of the parallelogram

$X+4+X+4=22$

$2X+8=22$ /-8

$2X=14$ /:2

$X=7$

First step of the answer:

$CD=7$

Area $ABCD=CD\cdot LD$ (height)

Area $ABCD=7\cdot3$

Area $ABCD = 21$

Answer:

Area of the parallelogram: $ABCD=21 cm²$

Consignment

Given the parallelogram $ABCD$

The area of the parallelogram is $98\operatorname{cm}²$

$\frac{AE}{DC}=\frac{1}{2}$

Objective:

Find a $DC$

Solution

According to the existing data we can calculate a $AE$

$AE=\frac{1}{2}DC$

$\text{ABCD}=DC\cdot AE=$

We replace the data accordingly

$98=\frac{1}{2}DC\cdot DC$

Multiply by $2$

$196=DC^2$

Take the root

$DC=14$

Answer

$14$

Reference

The area of the parallelogram $ABCD$ is $72\operatorname{cm}²$

Find a $DC$

Solution

$AE$ is the external height $DC$

$\text{ABCD}=DC\cdot AE=$

Replace the data accordingly

$72y=DC\cdot9$

Divide by $9$

$\frac{72y}{9}=DC$

$8y=DC$

Answer

$8y$

Consignment

Given the parallelogram $ABCD$

The relationship between $AE$ and $DC$ is $4:7$

Find the area of the parallelogram $ABCD$

Solution

According to the existing data we first calculate a $DC$

$\frac{AE}{DC}=\frac{4}{7}$

Replace a $AE$

$\frac{8}{DC}=\frac{4}{7}$

Multiply by cross

$8\cdot7=4\cdot DC$

Divide by $4$

$DC=\frac{8\cdot7}{4}=7\cdot2=14$

$\text{ABCD}=DC\cdot AE=$

Replace accordingly

$8\cdot14=112$

Answer

$112$

Assignment

Given the parallelogram of the figure

Its area is equal to $40\operatorname{cm}²$

Find a $AE$

Solution

$\text{ABCD}=DC\cdot AE=$

$DC=AB=8$

In the parallelogram the opposite sides are equal to each other.

Replace the data accordingly

$40=AE\cdot8$

Divide by $8$

$AE=5$

Answer

$5$

Consignment

Given the parallelogram $ABCD$

Its area is equal to $100\operatorname{cm}²$

Find a $AD$

Solution

$\text{ABCD}=DC\cdot AE=$

Replace the data accordingly

$100=6\cdot AD$

Divide by $6$

$AD=16.67$

Answer

$16.67$