Examples with solutions for Area of a Parallelogram
Exercise #1
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 #2
Calculate the area of the parallelogram according to the data in the diagram.
Video Solution
Step-by-Step Solution
Since 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 #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 with a perimeter of 38 cm.
AB is twice as long as CE.
AD is three times shorter than CE.
CE is the height of the parallelogram.
Calculate the area of the parallelogram.
Video Solution
Step-by-Step Solution
Let's call CE as X
According to the data
AB=x+2,AD=x−3
The perimeter of the parallelogram:
2(AB+AD)
38=2(x+2+x−3)
38=2(2x−1)
38=4x−2
38+2=4x
40=4x
x=10
Now it can be argued:
AD=10−3=7,CE=10
The area of the parallelogram:
CE×AD=10×7=70
Answer
70 cm²
Exercise #12
The area of trapezoid ABCD is X cm².
The line AE creates triangle AED and parallelogram ABCE.
The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.
Calculate the ratio between sides DE and EC.
Video Solution
Step-by-Step Solution
To calculate the ratio between the sides we will use the existing figure:
AABCEAAED=31
We calculate the ratio between the sides according to the formula to find the area and then replace the data.
We know that the area of triangle ADE is equal to:
AADE=2h×DE
We know that the area of the parallelogram is equal to:
AABCD=h×EC
We replace the data in the formula given by the ratio between the areas:
h×EC21h×DE=31
We solve by cross multiplying and obtain the formula:
h×EC=3(21h×DE)
We open the parentheses accordingly:
h×EC=1.5h×DE
We divide both sides by h:
EC=h1.5h×DE
We simplify to h:
EC=1.5DE
Therefore, the ratio between is: DEEC=1.51
Answer
1:1.5
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: