Examples with solutions for Area of a Parallelogram: Using Pythagoras' theorem

Exercise #1

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

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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

Exercise #2

The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.

AE = 8

BC = 5

P=24P=24P=24555AAABBBCCCDDDEEEFFF8

What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

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:

2×8+2X=24 2\times8+2X=24

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

We isolate X:

2X=8 2X=8

and divide by 2:

X=4 X=4

Now we can use the Pythagorean theorem to find 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 isolate the variable

EB2=9 EB^2=9

We take the square root of the equation.

EB=3 EB=3

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

AB= AE+EB AB=\text{ AE}+EB

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

And therefore we will apply the area formula:

11×4=44 11\times4=44

Answer

44

Exercise #3

Look at the parallelograms ABCD and BCED below.

BD=CD=5 BD=CD=5

BC=6 BC=6

Calculate the area of the parallelogram BCAE.

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Video Solution

Answer

24

Exercise #4

Calculate the area of the parallelogram ABCD according to the following data:

AD=12 AD=12

DC=7 DC=7

CB=8 CB=8

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Video Solution

Answer

68.937

Exercise #5

ABCD is a square with a side length of 8 cm.

EB = 10

What is the area of the parallelogram EBFC?

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Video Solution

Answer

112 cm²

Exercise #6

The rectangle ABCD and parallelogram EBFD are shown below.

BF = 5

DC = 10

EB = 7

What is the area of the parallelogram EBFD?

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Video Solution

Answer

28 cm²