Examples with solutions for Area of a Parallelogram
Exercise #1
Calculate the area of the parallelogram according to the data in the diagram.
Video Solution
Step-by-Step Solution
We know that ABCD is a parallelogram. According to the properties of parallelograms, each pair of opposite sides are equal and parallel.
Therefore: CD=AB=10
We will calculate the area of the parallelogram using the formula of side multiplied by the height drawn from that side, so the area of the parallelogram is equal to:
SABCD=10×7=70cm2
Answer
70
Exercise #2
Calculate the area of the following parallelogram:
Video Solution
Step-by-Step Solution
To solve the exercise, we need to remember the formula for the area of a parallelogram:
Side * Height perpendicular to the side
We can identify that in the diagram, although it's not presented in the way we're familiar with, we are given the two essential pieces of information -
Side = 6
Height = 5
Let's substitute into the formula and calculate:
6*5=30
And that's the solution!
Answer
30 cm²
Exercise #3
Look at the parallelogram in the figure.
Its area is equal to 70 cm².
Calculate DC.
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 cm
Exercise #4
ABCD is a parallelogram.
Its perimeter is 47 cm.
What is its area?
Video Solution
Step-by-Step Solution
First, let's remember that the perimeter of a parallelogram is the sum of its sides,
which is
AB+BC+CD+DA
We recall that in a parallelogram, opposite sides are equal, so BC=AD=6
Let's substitute in the formula:
2AB+12=47
2AB=35
AB=17.5
Now, after finding the missing sides, we can continue to calculate the area.
Remember, the area of a parallelogram is side*height to the side.
17.5*8= 140
Answer
140 cm²
Exercise #5
Look at the parallelogram in the figure below.
Its area is equal to 40 cm².
Calculate AE.
Video Solution
Step-by-Step Solution
We are told that ABCD is a parallelogram,AB=CD=8According to the properties of a parallelogram, each pair of opposite sides are equal and parallel.
Hence to find AE we will need to use the area given to us in the formula in order to determine the area of the parallelogram:
S=DC×AE
40=8×AE
We divide both sides of the equation by 8:
8AE:8=40:8
AE=5
Answer
5 cm
Question 1
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
ABCD is a parallelogram whose perimeter is equal to 24 cm.
The side of the parallelogram is two times greater than the adjacent side (AB>AD).
CE is the height of the side AB
The area of the parallelogram is 24 cm².
Find the height of CE
Video Solution
Step-by-Step Solution
The perimeter of the parallelogram is calculated as follows:
SABCD=AB+BC+CD+DA Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC
According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued thatAB=DC=2BC
We inut the data we know in the formula to calculate the perimeter:
PABCD=2BC+BC+2BC+BC
We replace the given perimeter in the formula and add up all the BC coefficients accordingly:
24=6BC
We divide the two sections by 6
24:6=6BC:6
BC=4
We know thatAB=DC=2BCWe replace the data we obtained (BC=4)
AB=DC=2×4=8
As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4
To find EC we use the formula:AABCD=AB×EC
We replace the existing data:
24=8×EC
We divide the two sections by 824:8=8EC:8
3=EC
Answer
3 cm
Exercise #12
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?
Video Solution
Step-by-Step Solution
Let's first calculate the sides of the rectangle:
AEDF=AE×ED
Let's input the known data:
35=7×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×ED
Let's input the known data:
ABCD=9×5=45
Answer
45 cm².
Exercise #13
The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.
AE = 8
BC = 5
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
16+2X=24
We isolate X:
2X=8
and divide by 2:
X=4
Now we can use the Pythagorean theorem to find EB.
(Pythagoras: A2+B2=C2)
EB2+42=52
EB2+16=25
We isolate the variable
EB2=9
We take the square root of the equation.
EB=3
The area of a parallelogram is the height multiplied by the side to which the height descends, that isAB×EC.
AB= AE+EB
AB=8+3=11
And therefore we will apply the area formula:
11×4=44
Answer
44
Exercise #14
ABCD is a parallelogram.
Angle ACB is equal to angle EBC.
BF = 6
CE = 9
BF is perpendicular to DE.
Calculate the area of the parallelogram.
Video Solution
Step-by-Step Solution
Given that angle ACB is equal to angle CBE, it follows that AC is parallel to BE
since alternate angles between parallel lines are equal.
As we know that ABCD is a parallelogram, AB is parallel to DC and therefore AB is also parallel to CE since it is a line that continues DC.
Given that AC is parallel to BE and, in addition, AB is parallel to CE, it can be argued that ABCE is a parallelogram and, therefore, each pair of opposite sides in a parallelogram are parallel and equal.
From this it is concluded that AB=CE=9
Now we calculate the area of the parallelogram ABCD according to the data.
SABCD=AB×BF
We replace the data accordingly:
SABCD=9×6=54
Answer
54 cm²
Exercise #15
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?
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 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:2πR We replace and solve:
2πR=25.13 πR=12.565 R≈4
The height of the parallelogram is equal to two radii, that is, 8.
And from here you can calculate with a parallelogram area formula: