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

🏆Practice area of a parallelogram

How is the area of a parallelogram calculated?

We can calculate the area of a parallelogram by multiplying one of its sides by its relative height.

To understand it better, we can use the following figure and the accompanying formula:

A=DC×H1=BC×H2 A=DC\times H1=BC\times H2

It can be seen that: H1 H1 and H2 H2 are the two heights corresponding to the bases DC DC and BC BC respectively.

Area of a Parallelogram

1 - Area of a Parallelogram

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Test yourself on area of a parallelogram!

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A parallelogram has a length equal to 6 cm and a height equal to 4.5 cm.

Calculate the area of the parallelogram.

6664.54.54.5

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What is a parallelogram?

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

Property of parallelograms

A1 - Parallelogram KLMN

  • 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 180o 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:

KM2+LN2=KL2+LM2+MN2+NK2 KM^{2}+LN^{2}=KL^{2}+LM^{2}+MN^{2}+NK^{2}

Or, in other words:

KM2+LN2=2KL2+2LM2 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 90o 90^o degrees and its two diagonals have the same length.

Rectangle

R - 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

5 - 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

6 - Square

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Practice exercises for finding the area of a parallelogram

Exercise 1

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

  • MN=10cm MN=10cm
  • KP=5cm KP=5cm

Area of a Parallelogram

Exercise 1

A7 - Area of a Parallelogram

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=MNKP=105=50cm2 A=MN\cdot KP=10\cdot5=50cm²

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


Exercise 2

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

cm - Area_of_a_parallelogram_-Exercise_2

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 DC and consider as its relative height AS AS ; the other way, is to consider the adjacent side BC BC as the base and its relative height AF 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=DCAS=93=27 A=DC\cdot AS=9\cdot3=27

A=BCAF=65=30 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?

Exercise 3

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

  • DE=12cm DE=12\operatorname{cm}
  • KG=5cm KG=5\operatorname{cm}
  • DK=9cm DK=9\operatorname{cm}
A9 - Area_of_a_parallelogram_-_Exercise_3

Solution:

If we look at the illustration, we see that DK DK refers to the external height of the parallelogram DEFG 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=12DE= GF=12 and DE DE parallel to GF GF .

To calculate the area of this parallelogram we do not need the data about the length of KG 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=GFDK=129=108cm2 A=GF\cdot DK=12\cdot9=108cm²

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


Additional exercises

Exercise 4

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

AE=8 AE = 8

the AEFC rectangle with a perimeter of 24

Task:

What is the area of the parallelogram?

Solution:

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

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

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

We substitute the known data:

2×8+2X=24 2\times8+2X=24

16+2X=24 16+2X=24

We clear the X X

2X=8 2X=8

And divide by 2 2

X=4 X=4

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

Pythagoras -A2+B2=C2 A^2+B^2=C^2

EB2+42=52 EB^2+4^2=5^2

EB2+16=25 EB^2+16=25

We clear EB EB

EB2=9 EB^2=9

Calculate the root

EB=3 EB=3

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

AB= AE+EB AB=\text{ AE}+EB

On the other hand,

AB=8+3=11 AB=8+3=11

Substitute the data into the area formula:

11×4=44 11\times4=44

Answer: 44 44


Check your understanding

Exercise 5

Given that:

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

AC=4cm AC = 4cm

The height =? =?

And the side of the parallelogram

DL=3cm DL = 3cm

DL = 3cm and the perimeter of the parallelogram ABCD is equal to 22 cm

Task:

Calculate the area of the parallelogram. ABCD ABCD

Solution:

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

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

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

X+4+X+4=22 X+4+X+4=22

2X+8=22 2X+8=22 /-8

2X=14 2X=14 /:2

X=7 X=7

First step of the answer:

CD=7 CD=7

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

Area ABCD=73 ABCD=7\cdot3

Area ABCD=21 ABCD = 21

Answer:

Area of the parallelogram: ABCD=21cm2 ABCD=21 cm²


Exercise 6

Consignment

Given the parallelogram ABCD ABCD

The area of the parallelogram is 98cm2 98\operatorname{cm}²

AEDC=12 \frac{AE}{DC}=\frac{1}{2}

Objective:

Find a DC DC

Exercise 6- Task Given the parallelogram ABCD

Solution

According to the existing data we can calculate a AE AE

AE=12DC AE=\frac{1}{2}DC

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

We replace the data accordingly

98=12DCDC 98=\frac{1}{2}DC\cdot DC

Multiply by 2 2

196=DC2 196=DC^2

Take the root

DC=14 DC=14

Answer

14 14


Do you think you will be able to solve it?

Exercise 7

Reference

The area of the parallelogram ABCD ABCD is 72cm2 72\operatorname{cm}²

Find a DC DC

Exercise 7-- Assignment The area of the parallelogram

Solution

AE AE is the external height DC DC

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

Replace the data accordingly

72y=DC9 72y=DC\cdot9

Divide by 9 9

72y9=DC \frac{72y}{9}=DC

8y=DC 8y=DC

Answer

8y 8y


Exercise 8

Consignment

Given the parallelogram ABCD ABCD

The relationship between AEAE and DCDC is 4:74:7

Find the area of the parallelogram ABCD ABCD

Exercise 8 Task Given the parallelogram ABCD

Solution

According to the existing data we first calculate a DC DC

AEDC=47 \frac{AE}{DC}=\frac{4}{7}

Replace a AE AE

8DC=47 \frac{8}{DC}=\frac{4}{7}

Multiply by cross

87=4DC 8\cdot7=4\cdot DC

Divide by 4 4

DC=874=72=14 DC=\frac{8\cdot7}{4}=7\cdot2=14

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

Replace accordingly

814=112 8\cdot14=112

Answer

112 112


Test your knowledge

Exercise 9

Assignment

Given the parallelogram of the figure

Its area is equal to 40cm2 40\operatorname{cm}²

Find a AE AE

Exercise 9 Assignment Given the parallelogram in the figure

Solution

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

DC=AB=8 DC=AB=8

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

Replace the data accordingly

40=AE8 40=AE\cdot8

Divide by 8 8

AE=5 AE=5

Answer

5 5


Exercise 10

Consignment

Given the parallelogram ABCD ABCD

Its area is equal to 100cm2 100\operatorname{cm}²

Find a AD AD

Exercise 10 Task Given the parallelogram ABCD

Solution

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

Replace the data accordingly

100=6AD 100=6\cdot AD

Divide by 6 6

AD=16.67 AD=16.67

Answer

16.67 16.67


Do you know what the answer is?

examples with solutions for area of a parallelogram

Exercise #1

Look at the parallelogram in the figure.

Its area is equal to 70 cm².

Calculate DC.

555AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

The formula for the area of a parallelogram:

Height * The side to which the height descends.

We replace in the formula all the known data, including the area:

5*DC = 70

We divide by 5:

DC = 70/5 = 14

And that's how we reveal the unknown!

Answer

14 14 cm

Exercise #2

Look at the parallelogram in the figure below.

Its area is equal to 40 cm².

Calculate AE.

888AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Given that ABCD is a parallelogram,AB=CD=8 AB=CD=8 According to the properties of the parallelogram, each pair of opposite sides are equal and parallel.

To find AE we will use the area given to us in the formula to find the area of the parallelogram:

S=DC×AE S=DC\times AE

40=8×AE 40=8\times AE

We divide both sides of the equation by 8:

8AE:8=40:8 8AE:8=40:8

AE=5 AE=5

Answer

5 5 cm

Exercise #3

ABCD parallelogram, it is known that:

BE is perpendicular to DE

BF is perpendicular to DF

BF=8 BE=4 AD=6 DC=12

Calculate the area of the parallelogram in 2 different ways

121212666444888AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

In this exercise, we are given two heights and two sides.

It is important to keep in mind: The external height can also be used to calculate the area

Therefore, we can perform the operation of the following exercise:

The height BF * the side AD

8*6

 

The height BE the side DC
4
*12

 The solution of these two exercises is 48, which is the area of the parallelogram.

 

Answer

48 cm²

Exercise #4

AE is the height of the parallelogram ABCD.

AB is 3 cm longer than AE.

The area of ABCD is 32 cm².

Calculate the length of side AB.

S=32S=32S=32AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Keep in mind that AB is 3 cm greater than AE, so we must pay attention to the data when we put the formula to calculate the parallelogram:

Height multiplied by the side of the height:

AB×AE=S AB\times AE=S

We will mark AE with the letter a and therefore AB will be a+3:

a×(a+3)=32 a\times(a+3)=32

We open the parentheses:

a2+3a=32 a^2+3a=32

We use the trinomial/roots formula:

a2+3a32=0 a^2+3a-32=0 (a+8)(a5)=0 (a+8)(a-5)=0

That means we have two options:

a=8,a=5 a=-8,a=5

Since it is not possible to place a negative side in the formula to calculate the areaa=5 a=5

Now we can calculate the sides:

AE=5 AE=5

AB=5+3=8 AB=5+3=8

Answer

8 cm

Exercise #5

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

777555AAABBBCCCDDDEEE2

What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

To find the area,

first, the height of the parallelogram must be found.

To conclude, let's take a look at triangle EBC.

Since we know it is a right triangle (since it is the height of the parallelogram)

the Pythagorean theorem can be used:

a2+b2=c2 a^2+b^2=c^2

In this case: EB2+EC2=BC2 EB^2+EC^2=BC^2

We place the given information: 22+EC2=52 2^2+EC^2=5^2

We isolate the variable:EC2=52+22 EC^2=5^2+2^2

We solve:EC2=254=21 EC^2=25-4=21

EC=21 EC=\sqrt{21}

Now all that remains is to calculate the area.

It is important to remember that for this, the length of each side must be used.
That is, AE+EB=2+7=9

21×9=41.24 \sqrt{21}\times9=41.24

Answer

41.24

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