Area of a Parallelogram: Using additional geometric shapes

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?

Let's first calculate the sides of the rectangle:

$AEDF=AE\times ED$

Let's input the known data:

$35=7\times ED$

Let's divide the two legs by 7:

$ED=5$

Since AEDF is a rectangle, we can claim that:

ED=FD=7

Let's calculate side CD:

$2+7=9$

Let's calculate the area of parallelogram ABCD:

$ABCD=CD\times ED$

Let's input the known data:

$ABCD=9\times5=45$

45 cm².

Below is a circle bounded by a parallelogram:

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

What is the area of the parallelogram?

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:

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

And from here we can calculate:

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

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:$2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

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

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

$AlturaXLado$

$9\times8\approx72$

$\approx72$

The parallelogram ABCD is shown below.

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

ECFD is a rhombus whose area is 24 cm².

What is the area of ABCD?

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 $area=DC\times h_{DC}$

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:

$area=BC\times h_{BC}$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.

It is not possible to calculate.

ABCD is a parallelogram
BFCE is a deltoid

What is the area of the parallelogram ABCD?

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

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 (

$A^2+B^2=C^2$)

$BG^2+4^2=5^2$

$BG^2+16=25$

$BG^2=9$

$BG=3$

Now, let's start looking for GC.

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

Now we can also do Pythagoras in the triangle GCE.

$GC^2+4^2=9^2$

$GC^2+16=81$

$GC^2=65$

$GC=\sqrt{65}$

Now we can calculate the side BC:

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

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:

$\frac{HD}{BG}=\frac{HC}{GE}$

$\frac{HD}{BG}=\frac{7.5}{3}=2.5$

$\frac{HC}{EG}=\frac{HC}{4}=2.5$

$HC=10$

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

$10\times11\approx110$

$\approx110$

The following is a circle enclosed in a parallelogram:

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

What is the area of the zones marked in blue?

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:

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

From here we can calculate:

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

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:$2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

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:

$\text{Lado }x\text{ Altura}$$9\times8\approx72$

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

$\pi4^2=50.26$

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

$72-56.24\approx21.73$

$\approx21.73$

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

EB = 10

What is the area of the parallelogram EBFC?

112 cm²

The rectangle ABCD and parallelogram EBFD are shown below.

BF = 5

DC = 10

EB = 7

What is the area of the parallelogram EBFD?

28 cm²

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.

32 cm²

ABCD is a parallelogram.

The triangle BEC is equilateral.

What is the area of the parallelogram?

80 cm²

Look at the parallelogram ABCD in the figure below.

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

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

What is the area of the parallelogram?

$30$ cm²

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?

$24x$ cm²

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

Calculate the area of the parallelogram ABED.

$9x$ cm²

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

BO is the radius.

ABCD is a parallelogram.
BO is perpendicular to DC.

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

What is the area of the parallelogram?

$72\frac{a}{\pi}$ cm²