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:

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

Examples of parallelograms

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

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Practice exercises for finding the area of a parallelogram

Exercise 1

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 $50cm²$.

Exercise 2

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.

Do you know what the answer is?

Question 1

Calculate the area of the parallelogram using the data in the figure:

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 $108 cm²$.

Additional exercises

Exercise 4

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 $EB$.