What is a parallelogram? The parallelogram is a four-sided polygon (quadrilateral), whose opposite sides are. Property of parallelograms
The opposite angles of the parallelogram have the same size. The opposite sides of the parallelogram have the same length. Parallelograms have two intersecting diagonals that create two pairs of triangles. In addition, the four triangles that are formed have the same area. The angles of the parallelogram complement each other until they reach 18 0 o 180^o 18 0 o degrees. The sum of the squares of its diagonals is equal to the sum of the squares of the four sides of the parallelogram. In other words:
K M 2 + L N 2 = K L 2 + L M 2 + M N 2 + N K 2 KM^{2}+LN^{2}=KL^{2}+LM^{2}+MN^{2}+NK^{2} K M 2 + L N 2 = K L 2 + L M 2 + M N 2 + N K 2
Or, in other words:
K M 2 + L N 2 = 2 K L 2 + 2 L M 2 KM^{2}+LN^{2}=2KL^{2}+2LM^{2} K M 2 + L N 2 = 2 K L 2 + 2 L M 2
Examples of parallelograms Rectangle : is a parallelogram in which all its angles are right angles, that is, they measure 9 0 o 90^o 9 0 o degrees and its two diagonals have the same length.Rectangle
Rhombus : a parallelogram whose four sides are of equal length (and its two diagonals intersect at right angles, that is, they are perpendicular).Rhombus
Square : is a parallelogram that meets the definition of rectangle and rhombus (but also its two diagonals are perpendicular and have the same length).Square
Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge
Question 2 a It is not possible to calculate.
Correct Answer: It is not possible to calculate.
Practice exercises for finding the area of a parallelogram Exercise 1 Find the area of the parallelogram K L M N KLMN K L MN illustrated in the figure below using the data provided:
M N = 10 c m MN=10cm MN = 10 c m K P = 5 c m KP=5cm K P = 5 c m Area of a Parallelogram
Exercise 1
Solution :
This is a fairly simple exercise in which we must substitute the given data in the formula corresponding to the area of a parallelogram:
A = M N ⋅ K P = 10 ⋅ 5 = 50 c m 2 A=MN\cdot KP=10\cdot5=50cm² A = MN ⋅ K P = 10 ⋅ 5 = 50 c m 2
Answer : The area of the parallelogram K L M N KLMN K L MN is 50 c m 2 50cm² 50 c m 2 .
Exercise 2 Analyze the illustration below and indicate if there are any errors in the data given. Explain your answer.
Solution :
This exercise deals with the area of a parallelogram. As we have already said, the area of this geometric shape can be calculated in two ways. With the first one, we must use as base the side D C DC D C and consider as its relative height A S AS A S ; the other way, is to consider the adjacent side B C BC BC as the base and its relative height A F AF A F . The answer we obtain by applying both methods must be the same.
We substitute the data in the formula and we obtain the following:
A = D C ⋅ A S = 9 ⋅ 3 = 27 A=DC\cdot AS=9\cdot3=27 A = D C ⋅ A S = 9 ⋅ 3 = 27
A = B C ⋅ A F = 6 ⋅ 5 = 30 A=BC\cdot AF=6\cdot5=30 A = BC ⋅ A F = 6 ⋅ 5 = 30
As we can see, we have obtained a different result by applying one or the other method and, therefore, the given data are wrong.
Do you know what the answer is?
Question 1 a It is not possible to calculate.
Correct Answer: It is not possible to calculate.
Exercise 3 Find the area of the parallelogram D E F G DEFG D EFG according to the illustration and the data below:
D E = 12 cm DE=12\operatorname{cm} D E = 12 cm K G = 5 cm KG=5\operatorname{cm} K G = 5 cm D K = 9 cm DK=9\operatorname{cm} DK = 9 cm Solution:
If we look at the illustration, we see that D K DK DK refers to the external height of the parallelogram D E F G DEFG D EFG .
According to the characteristics of the parallelogram that we have just learned, the opposite sides of a parallelogram are identical and parallel to each other, that is: D E = G F = 12 DE= GF=12 D E = GF = 12 and D E DE D E parallel to G F GF GF .
To calculate the area of this parallelogram we do not need the data about the length of K G KG K G since this information is not useful for such a calculation, but was given to us only to confuse us. To calculate the area of a parallelogram, we only need the length of a side and its relative height.
That said, we substitute the data into the formula and we will get the following:
A = G F ⋅ D K = 12 ⋅ 9 = 108 c m 2 A=GF\cdot DK=12\cdot9=108cm² A = GF ⋅ DK = 12 ⋅ 9 = 108 c m 2
Answer : The area of the parallelogram D E F G DEFG D EFG is 108 c m 2 108 cm² 108 c m 2 .
Additional exercises Exercise 4 Inside the parallelogram A B C D ABCD A BC D is the rectangle A E C F AECF A ECF with a perimeter of 24 24 24 .
A E = 8 AE = 8 A E = 8
Task:
What is the area of the parallelogram?
Solution:
In the first step we must find the length E C EC EC , which we will identify as X X X .
We know that the perimeter of the rectangle is equal to the sum of its sides ( A E + E C + C F + F A ) (AE+EC+CF+FA) ( A E + EC + CF + F A ) .
Because in the rectangle the opposite sides are equal, we can write the formula like this: 2 A E + 2 E C = 24 2AE+2EC=24 2 A E + 2 EC = 24
We substitute the known data:
2 × 8 + 2 X = 24 2\times8+2X=24 2 × 8 + 2 X = 24
16 + 2 X = 24 16+2X=24 16 + 2 X = 24
We clear the X X X
2 X = 8 2X=8 2 X = 8
And divide by 2 2 2
X = 4 X=4 X = 4
Now, we can use the Pythagorean formula to calculate E B EB EB .
Pythagoras -A 2 + B 2 = C 2 A^2+B^2=C^2 A 2 + B 2 = C 2
E B 2 + 4 2 = 5 2 EB^2+4^2=5^2 E B 2 + 4 2 = 5 2
E B 2 + 16 = 25 EB^2+16=25 E B 2 + 16 = 25
We clear E B EB EB
E B 2 = 9 EB^2=9 E B 2 = 9
Calculate the root
E B = 3 EB=3 EB = 3
The area of the parallelogram is the product of the side A B AB A B by its relative height EC A B × E C AB\times EC A B × EC
A B = AE + E B AB=\text{ AE}+EB A B = AE + EB
On the other hand,
A B = 8 + 3 = 11 AB=8+3=11 A B = 8 + 3 = 11
Substitute the data into the area formula:
11 × 4 = 44 11\times4=44 11 × 4 = 44
Answer : 44 44 44
Check your understanding
Question 1 b It is not possible to calculate.
Correct Answer: It is not possible to calculate.
Exercise 5 Given that:
The perimeter of the parallelogram A B C D ABCD A BC D is equal to 22 c m 22 cm 22 c m . D L = 3 c m DL = 3cm D L = 3 c m
A C = 4 c m AC = 4cm A C = 4 c m
The height = ? =? = ?
And the side of the parallelogram
D L = 3 c m DL = 3cm D L = 3 c m
Task:
Calculate the area of the parallelogram. A B C D ABCD A BC D
Solution:
Parallel opposite sides are equal A C = B D = 4 c m AC=BD=4cm A C = B D = 4 c m
Parallel opposite sides are equal A B = C D = X c m AB=CD=Xcm A B = C D = X c m
A B + B D + C D + A C = AB+BD+CD+AC\text{ }= A B + B D + C D + A C = Perimeter of the parallelogram
X + 4 + X + 4 = 22 X+4+X+4=22 X + 4 + X + 4 = 22
2 X + 8 = 22 2X+8=22 2 X + 8 = 22 /-8
2 X = 14 2X=14 2 X = 14 /:2
X = 7 X=7 X = 7
First step of the answer:
C D = 7 CD=7 C D = 7
Area A B C D = C D ⋅ L D ABCD=CD\cdot LD A BC D = C D ⋅ L D (height)
Area A B C D = 7 ⋅ 3 ABCD=7\cdot3 A BC D = 7 ⋅ 3
Area A B C D = 21 ABCD = 21 A BC D = 21
Answer:
Area of the parallelogram: A B C D = 21 c m 2 ABCD=21 cm² A BC D = 21 c m 2
Exercise 6 Consignment
Given the parallelogram A B C D ABCD A BC D
The area of the parallelogram is 98 cm 2 98\operatorname{cm}² 98 cm 2
A E D C = 1 2 \frac{AE}{DC}=\frac{1}{2} D C A E = 2 1
Objective:
Find a D C DC D C
Solution
According to the existing data we can calculate a A E AE A E
A E = 1 2 D C AE=\frac{1}{2}DC A E = 2 1 D C
ABCD = D C ⋅ A E = \text{ABCD}=DC\cdot AE= ABCD = D C ⋅ A E =
We replace the data accordingly
98 = 1 2 D C ⋅ D C 98=\frac{1}{2}DC\cdot DC 98 = 2 1 D C ⋅ D C
Multiply by 2 2 2
196 = D C 2 196=DC^2 196 = D C 2
Take the root
D C = 14 DC=14 D C = 14
Answer
14 14 14
Do you think you will be able to solve it?
Exercise 7 Reference
The area of the parallelogram A B C D ABCD A BC D is 72 cm 2 72\operatorname{cm}² 72 cm 2
Find a D C DC D C
Solution
A E AE A E is the external height D C DC D C
ABCD = D C ⋅ A E = \text{ABCD}=DC\cdot AE= ABCD = D C ⋅ A E =
Replace the data accordingly
72 y = D C ⋅ 9 72y=DC\cdot9 72 y = D C ⋅ 9
Divide by 9 9 9
72 y 9 = D C \frac{72y}{9}=DC 9 72 y = D C
8 y = D C 8y=DC 8 y = D C
Answer
8 y 8y 8 y
Exercise 8 Consignment
Given the parallelogram A B C D ABCD A BC D
The relationship between A E AE A E and D C DC D C is 4 : 7 4:7 4 : 7
Find the area of the parallelogram A B C D ABCD A BC D
Solution
According to the existing data we first calculate a D C DC D C
A E D C = 4 7 \frac{AE}{DC}=\frac{4}{7} D C A E = 7 4
Replace a A E AE A E
8 D C = 4 7 \frac{8}{DC}=\frac{4}{7} D C 8 = 7 4
Multiply by cross
8 ⋅ 7 = 4 ⋅ D C 8\cdot7=4\cdot DC 8 ⋅ 7 = 4 ⋅ D C
Divide by 4 4 4
D C = 8 ⋅ 7 4 = 7 ⋅ 2 = 14 DC=\frac{8\cdot7}{4}=7\cdot2=14 D C = 4 8 ⋅ 7 = 7 ⋅ 2 = 14
ABCD = D C ⋅ A E = \text{ABCD}=DC\cdot AE= ABCD = D C ⋅ A E =
Replace accordingly
8 ⋅ 14 = 112 8\cdot14=112 8 ⋅ 14 = 112
Answer
112 112 112
Test your knowledge
Question 3 a It is not possible to calculate.
Correct Answer: It is not possible to calculate.
Exercise 9 Assignment
Given the parallelogram of the figure
Its area is equal to 40 cm 2 40\operatorname{cm}² 40 cm 2
Find a A E AE A E
Solution
ABCD = D C ⋅ A E = \text{ABCD}=DC\cdot AE= ABCD = D C ⋅ A E =
D C = A B = 8 DC=AB=8 D C = A B = 8
In the parallelogram the opposite sides are equal to each other.
Replace the data accordingly
40 = A E ⋅ 8 40=AE\cdot8 40 = A E ⋅ 8
Divide by 8 8 8
A E = 5 AE=5 A E = 5
Answer
5 5 5
Exercise 10 Consignment
Given the parallelogram A B C D ABCD A BC D
Its area is equal to 100 cm 2 100\operatorname{cm}² 100 cm 2
Find a A D AD A D
Solution
ABCD = D C ⋅ A E = \text{ABCD}=DC\cdot AE= ABCD = D C ⋅ A E =
Replace the data accordingly
100 = 6 ⋅ A D 100=6\cdot AD 100 = 6 ⋅ A D
Divide by 6 6 6
A D = 16.67 AD=16.67 A D = 16.67
Answer
16.67 16.67 16.67
Do you know what the answer is?
Question 2 b It is not possible to calculate.
Correct Answer: It is not possible to calculate.
Examples with solutions for Area of a Parallelogram Exercise #1 Look at the parallelogram in the figure.
Its area is equal to 70 cm².
Calculate DC.
5 5 5 A A A B B B C C C D D D E E E
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 Exercise #2 Look at the parallelogram in the figure below.
Its area is equal to 40 cm².
Calculate AE.
8 8 8 A A A B B B C C C D D D E E E
Video Solution Step-by-Step Solution We are told that ABCD is a parallelogram,A B = C D = 8 AB=CD=8 A B = C D = 8 According 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 = D C × A E S=DC\times AE S = D C × A E
40 = 8 × A E 40=8\times AE 40 = 8 × A E
We divide both sides of the equation by 8:
8 A E : 8 = 40 : 8 8AE:8=40:8 8 A E : 8 = 40 : 8
A E = 5 AE=5 A E = 5
Answer Exercise #3 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
12 12 12 6 6 6 4 4 4 8 8 8 A A A B B B C C C D D D E E E F F F
Video Solution Step-by-Step Solution In this exercise, we are given two heights and two sides.
It is important to keep in mind: The external height can also be used to calculate the area
Therefore, we can perform the operation of the following exercise:
The height BF * the side AD
8*6
The height BE the side DC 4 *12
The solution of these two exercises is 48, which is the area of the parallelogram.
Answer Exercise #4 ABCD is a parallelogram.
CE is its height.
CB = 5 AE = 7 EB = 2
7 7 7 5 5 5 A A A B B B C C C D D D E E E 2
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:
a 2 + b 2 = c 2 a^2+b^2=c^2 a 2 + b 2 = c 2
In this case: E B 2 + E C 2 = B C 2 EB^2+EC^2=BC^2 E B 2 + E C 2 = B C 2
We place the given information: 2 2 + E C 2 = 5 2 2^2+EC^2=5^2 2 2 + E C 2 = 5 2
We isolate the variable:E C 2 = 5 2 + 2 2 EC^2=5^2+2^2 E C 2 = 5 2 + 2 2
We solve:E C 2 = 25 − 4 = 21 EC^2=25-4=21 E C 2 = 25 − 4 = 21
E C = 21 EC=\sqrt{21} EC = 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 21 × 9 = 41.24
Answer Exercise #5 AE is the height of the parallelogram ABCD. AB is 3 cm longer than AE. The area of ABCD is 32 cm². Calculate the length of side AB.
S=32 S=32 S=32 A A A B B B C C C D D D E E E
Video Solution Step-by-Step Solution Keep in mind that AB is 3 cm greater than AE, so we must pay attention to the data when we put the formula to calculate the parallelogram:
Height multiplied by the side of the height:
A B × A E = S AB\times AE=S A B × A E = S
We will mark AE with the letter a and therefore AB will be a+3:
a × ( a + 3 ) = 32 a\times(a+3)=32 a × ( a + 3 ) = 32
We open the parentheses:
a 2 + 3 a = 32 a^2+3a=32 a 2 + 3 a = 32
We use the trinomial/roots formula:
a 2 + 3 a − 32 = 0 a^2+3a-32=0 a 2 + 3 a − 32 = 0 ( a + 8 ) ( a − 5 ) = 0 (a+8)(a-5)=0 ( a + 8 ) ( a − 5 ) = 0
That means we have two options:
a = − 8 , a = 5 a=-8,a=5 a = − 8 , a = 5
Since it is not possible to place a negative side in the formula to calculate the areaa = 5 a=5 a = 5
Now we can calculate the sides:
A E = 5 AE=5 A E = 5
A B = 5 + 3 = 8 AB=5+3=8 A B = 5 + 3 = 8
Answer