Examples with solutions for Area of a Parallelogram: Using additional geometric shapes

Exercise #1

ABCD is a parallelogram and AEFD is a rectangle.

AE = 7

The area of AEFD is 35 cm².

CF = 2

What is the area of the parallelogram?

S=35S=35S=35777222AAAEEEDDDFFFCCCBBB

Video Solution

Step-by-Step Solution

Let's first calculate the sides of the rectangle:

AEDF=AE×ED AEDF=AE\times ED

Let's input the known data:

35=7×ED 35=7\times ED

Let's divide the two legs by 7:

ED=5 ED=5

Since AEDF is a rectangle, we can claim that:

ED=FD=7

Let's calculate side CD:

2+7=9 2+7=9

Let's calculate the area of parallelogram ABCD:

ABCD=CD×ED ABCD=CD\times ED

Let's input the known data:

ABCD=9×5=45 ABCD=9\times5=45

Answer

45 cm².

Exercise #2

Below is a circle bounded by a parallelogram:

36

All meeting points are tangential to the circle.
The circumference is 25.13.

What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

AE=AF=3 AE=AF=3
BG=BF=6 BG=BF=6

And from here we can calculate:

AB=AF+FB=3+6=9 AB=AF+FB=3+6=9

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

Since the circumference is 25.13.

Circumference formula:2πR 2\pi R
We replace and solve:

2πR=25.13 2\pi R=25.13
πR=12.565 \pi R=12.565
R4 R\approx4

The height of the parallelogram is equal to two radii, that is, 8.

And from here you can calculate with a parallelogram area formula:

AlturaXLado AlturaXLado

9×872 9\times8\approx72

Answer

72 \approx72

Exercise #3

The following is a circle enclosed in a parallelogram:

36

All meeting points are tangent to the circle.
The circumference is 25.13.

What is the area of the zones marked in blue?

Video Solution

Step-by-Step Solution

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

AE=AF=3 AE=AF=3
BG=BF=6 BG=BF=6

From here we can calculate:

AB=AF+FB=3+6=9 AB=AF+FB=3+6=9

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

It is known that the circumference of the circle is 25.13.

Formula of the circumference:2πR 2\pi R
We replace and solve:

2πR=25.13 2\pi R=25.13
πR=12.565 \pi R=12.565
R4 R\approx4

The height of the parallelogram is equal to two radii, that is, 8.

And from here it is possible to calculate the area of the parallelogram:

Lado x Altura \text{Lado }x\text{ Altura} 9×872 9\times8\approx72

Now, we calculate the area of the circle according to the formula:πR2 \pi R^2

π42=50.26 \pi4^2=50.26

Now, subtract the area of the circle from the surface of the trapezoid to get the answer:

7256.2421.73 72-56.24\approx21.73

Answer

21.73 \approx21.73

Exercise #4

The parallelogram ABCD is shown below.

BC is the diameter of the circle whose circumference is equal to 10π 10\pi cm.

ECFD is a rhombus whose area is 24 cm².

What is the area of ABCD?

333AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

Let's try to calculate the area in two ways.

In the first method, we will try to use the rhombus ECFD:

Let's try to calculate according to the formula area=DC×hDC area=DC\times h_{DC}

We will lower a height to DC and see that we do not have enough data to calculate, so we will not be able to calculate the area of the parallelogram using the rhombus.

In the second method , we will try to use the circle:

area=BC×hBC area=BC\times h_{BC} We will lower a height to BC and see that we do not have enough data to calculate, so we will not be able to calculate the area of the parallelogram using the circle.

From this it follows that we do not have enough data to calculate the area of parallelogram ABCD and therefore the exercise cannot be solved.

Answer

It is not possible to calculate.

Exercise #5

ABCD is a parallelogram
BFCE is a deltoid

555999444AAABBBCCCDDDFFFEEEHHHGGG7.5

What is the area of the parallelogram ABCD?

Video Solution

Step-by-Step Solution

First, we must remember the formula for the area of a parallelogram:Lado x Altura \text{Lado }x\text{ Altura} .

In this case, we will try to find the height CH and the side BC.

We start from the side

First, let's observe the small triangle EBG,

As it is a right triangle, we can use the Pythagorean theorem (

A2+B2=C2 A^2+B^2=C^2 )

BG2+42=52 BG^2+4^2=5^2

BG2+16=25 BG^2+16=25

BG2=9 BG^2=9

BG=3 BG=3

Now, let's start looking for GC.

First, remember that the deltoid has two pairs of equal adjacent sides, therefore:FC=EC=9 FC=EC=9

Now we can also do Pythagoras in the triangle GCE.

GC2+42=92 GC^2+4^2=9^2

GC2+16=81 GC^2+16=81

GC2=65 GC^2=65

GC=65 GC=\sqrt{65}

Now we can calculate the side BC:

BC=BG+GT=3+6511 BC=BG+GT=3+\sqrt{65}\approx11

Now, let's observe the triangle BGE and DHC

Angle BGE = 90°
Angle CHD = 90°
Angle CDH=EBG because these are opposite parallel angles.

Therefore, there is a ratio of similarity between the two triangles, so:

HDBG=HCGE \frac{HD}{BG}=\frac{HC}{GE}

HDBG=7.53=2.5 \frac{HD}{BG}=\frac{7.5}{3}=2.5

HCEG=HC4=2.5 \frac{HC}{EG}=\frac{HC}{4}=2.5

HC=10 HC=10

Now that there is a height and a side, all that remains is to calculate.

10×11110 10\times11\approx110

Answer

110 \approx110

Exercise #6

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

EB = 10

What is the area of the parallelogram EBFC?

101010AAABBBDDDCCCEEEFFF

Video Solution

Answer

112 cm²

Exercise #7

The parallelogram ABCD and the triangle BCE are shown below.

CE = 7
DE = 15

The area of the triangle BCE is equal to 14 cm².

Calculate the area of the parallelogram ABCD.

S=14S=14S=14777BBBCCCEEEAAADDD15

Video Solution

Answer

32 cm²

Exercise #8

The rectangle ABCD and parallelogram EBFD are shown below.

BF = 5

DC = 10

EB = 7

What is the area of the parallelogram EBFD?

101010555AAABBBCCCDDDEEEFFF7

Video Solution

Answer

28 cm²

Exercise #9

Look at the parallelogram ABCD in the figure below.

A semicircle with a length of 2.5π 2.5\pi cm is drawn as shown.

Another semicircle with an area of 4.5π 4.5\pi cm² is drawn on side DC.

What is the area of the parallelogram?

AAABBBCCCDDDEEE

Video Solution

Answer

30 30 cm²

Exercise #10

The parallelogram ABCD is shown below.

The area of the square GAEF is equal to 36 cm².

DC = 4X

What is the area of the parallelogram?

4X4X4XGGGBBBCCCFFFAAADDDEEES=36

Video Solution

Answer

24x 24x cm²

Exercise #11

ABCD is a trapezoid with an area of 10.5x 10.5x cm².

Calculate the area of the parallelogram ABED.

3X3X3XXXXAAABBBCCCDDDEEE

Video Solution

Answer

9x 9x cm²

Exercise #12

ABCD is a parallelogram.

The triangle BEC is equilateral.

What is the area of the parallelogram?

101010888AAABBBEEEDDDCCCFFF11

Video Solution

Answer

80 cm²

Exercise #13

The circumference of the circle in the diagram is 36a2 36a^2 cm.

BO is the radius.

ABCD is a parallelogram.
BO is perpendicular to DC.

DC = 4a \frac{4}{a}

What is the area of the parallelogram?

BBBOOOCCCDDDAAA

Video Solution

Answer

72aπ 72\frac{a}{\pi} cm²