Area of a trapezoid

🏆Practice trapezoid area

To find the area of a trapezoid, you need the following three pieces of information:

  • The length of base one
  • The length of base two
  • The height between the two bases

The formula to find the area of a trapezoid is as follows:

The sum of the bases multiplied by the height and then divided by two.

Formula of the trapezoid:

A=(Base 1+Base 2)×Height2 A=\frac{(Base~1+Base~2)\times Height}{2}

A1 - How do you calculate the area of a new trapezoid

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Test yourself on trapezoid area!

einstein

Given the following trapezoid:

AAABBBCCCDDD584

Calculate the area of the trapezoid ABCD.

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Geometric shapes, like the trapezoid, are all around us. Learning to work with these shapes will open doors for us in our studies as well as helping us to understand the world around us.

Let's start with finding the area of a trapezoid, one of the most important fundamental exercises.

Let's start at the beginning - What is a trapezoid?

A trapezoid is a quadrilateral that has one pair of parallel sides, which are called bases.

Its other two sides are opposite but not parallel.

There are several types of trapezoids, like the isosceles trapezoid (whose non-parallel sides have the same length, and their diagonals are equal), and the rectangular trapezoid (which has one side perpendicular to its bases).

Some quadrilaterals are often confused with the trapezoid, such as the parallelogram. However, the parallelogram has two pairs of parallel sides while the trapezoid has only one pair of parallel sides.


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Finding the area of a trapezoid

The formula to use to calculate the area of a trapezoid is as follows:

Base 1 (b1) plus base 2 (b2) times height (h) times two.isosceles, we know that

AFS=(B1+B2)×h2 AFS=\frac{(B1+B2)\times h}{2}

Now that you have reviewed the formula for finding the area of a trapezoid, here are three exercises to practice:

Example problems

Example 1 - area of an isosceles trapezoid

Suppose we have an isosceles trapezoid (whose non-parallel sides are equal) with the following values:

  • The length of the upper base that starts at the vertex A A and ends at the vertex B B is 6 cm 6~cm .
  • The length of the lower base starting at the vertex D D and ending at the vertex C C is 8 cm 8~cm .
  • The height, which is represented by the letter h h (from the word 'height'), is 4 cm 4~cm .

The sum of the two bases is equal to 6+8 6+8 .

Then, we multiply the sum by the height (4×14) (4\times14) , which is gives us 56 56 .

This is then divided by 2 2 , which is equal to 28 28 .

Therefore, the area of this trapezoid is equal to 28 square centimeters.

28=(6+8)×42 28=\frac{(6+8)\times 4}{2}

A2 - the area of this trapezoid is equal to 28 centimeters example 1 new


Do you know what the answer is?

Example 2 - area of a trapezoid

Suppose we have a trapezoid that is not isosceles (its non-parallel sides are not equal), and that has the following values:

  • The length of the base starting at vertex A A and ending at vertex B B is 6 cm 6~cm .
  • The length of the base starting at vertex D D and ending at vertex C C is 10 cm 10~cm .
  • The height (represented by the letter h h ) is 4 cm 4~cm .

The sum of the bases is

16 16 (6+10) (6+10) .

Then we multiply this sum by the height(16×4) \left(16\times4\right) , which equals 64 64 .

Finally, we divide 64 64 by 2 2 . We will get 32 32 .

Therefore, the area of the trapezoid is 32 cm2 32~cm² square centimeters.

32=(6+10)×42 32=\frac{(6+10)\times 4}{2}

A3 - trapezoid area is 32 square cm².


Example 3 - area of a rectangular trapezoid

Suppose we have a rectangular trapezoid (which has one side perpendicular to its bases) with the following values:

  • The length of the base starting at the vertex A A and ending at the vertex B B is 6 cm 6~cm .
  • The length of the base that starts at vertex D D and ends at vertex C C is 9 cm 9~cm .
  • The height between the bases (represented by the letter h) is 4 cm 4~cm .

First, we add the bases (6+9) \left(6+9\right) , which gives us 15 15 .
Then, we multiply 15 15 by the height (4×15) \left(4\times 15\right) , which equals 60 60 .
Finally, we divide 60 60 by 2 2 , which equals 30 30.
Therefore, the area of this trapezoid is equal to 30 cm 30~cm square centimeters. Given that the trapezoid is

30=(6+9)×42 30=\frac{(6+9)\times 4}{2}

A4 - trapezoid is equal to 30 square centimeters.

Although it's important to study with enthusiasm and motivation in order to be prepared for your exams, it is just as important to learn how to take breaks.
Whenever possible, take one day off a week from studying.
Giving yourself a moment to enjoy a well-deserved rest will renew your energy and motivation and make your studying more enjoyable and stress-free.

If you are preparing for an upcoming exam, try organizing your study time with specific goals and time slots.
This way, you will be able to track your progress as well as balance your study time efficiently.
Lastly, remember that when preparing for exams, not all students study in the same way. Every student needs to find the method of studying that is best suited for him or her.


Check your understanding

Additional exercises

Exercise 1

How do we calculate the area of a trapezoid?

Below we are given a trapezoid with the following values:

A5 - How do we calculate the area of a trapezoid

What is its height?

Solution

Formula for the area of a trapezoid:

(Base+Base)2×height \frac{(Base+Base)}{2}\times height

We don't have all our values for the formula, so we will have to work backwards to find the missing height.

9+62×h=30 \frac{9+6}{2}\times h=30

Now to reduce and solve:

152×h=30 \frac{15}{2}\times h=30

712×h=30 7\frac{1}{2}\times h=30

h=30152 h=\frac{30}{\frac{15}{2}}

h=6015 h=\frac{60}{15}

h=4 h=4

Answer Height BE BE is equal to 4 cm 4~cm .


Exercise 2

Given a rectangle ABCD ABCD formed by a trapezoid AKDC AKDC and a right triangle KBC KBC with the following measures:

DC=14 cm DC=14~cm

AD=5 cm AD=5~cm

KB=4 cm KB=4~cm

How many times is the area of the trapezoidAKCD AKCD greater than the area of the triangle KBC KBC ?

triangle KBC inside trapezium AKCD

DC=14 cm DC = 14~cm

AD=5 cm AD = 5~cm

KB=4 cm KB = 4~cm

To find out how many times the area of the trapezoid is greater than the area of the triangle, we will calculate the area of both figures and then divide the area of the trapezoid by the area of the triangle.

the two separate forms

To find the solution, we will need to calculate the area of the triangle and the area of the trapezoid.

The formula for the area of the triangle is as follows:

(Height×Base)2 \frac{\left(Height\times Base\right)}{2}

We know that the length of the base KB KB is 4 4 .

The height is CB CB .

Since the opposite sides of the rectangle are equal we know that AD=CB AD = CB .

Therefore CB=5 CB = 5 .

Now we can calculate

4×5=20 4\times 5=20

Finally we divide by two to get the area of the triangle.

202=10 \frac{20}{2}=10

Now we will calculate the area of the trapezoid:

Aˊrea=(AK+DC)×AD2 Área = \frac{\left(AK+D\text{C}\right) \times AD}{2}

We know the side DC=AB DC=AB because opposite sides in a rectangle are equal,

And the length of KB=4 KB =4 ,

Therefore we can calculate the length of AK AK .

ABKB=AK AB-KB=AK

144=10 14-4=10

Now we can substitute the values into the formula for the area of the trapezoid.

(14+10)×52 \frac{(14+10) \times 5}{2}

(24)×52 \frac{(24) \times 5}{2}

1202=60 \frac{120}{2} = 60

Now, all we have left to do is to divide the area of the trapezoid by the area of the triangle:

6010=6 \frac{60}{10}=6

Therefore, the trapezoid is six times greater than the triangle.

Answer:

The correct answer 6 6 times greater.


Do you think you will be able to solve it?

Exercise 3

Given an isosceles trapezoid ABCD ABCD

BC=7 cm BC = 7~cm

The height of the trapezoid is h=5 cm h = 5~cm

The perimeter of the trapezoid P=34 cm P = 34~cm

Calculate the area of the trapezoid

Exercise 3 Given an isosceles trapezium ABCD 5 cm

Solution:

To calculate the area of the trapezoid, we must analyze the given information.

Given that the trapezoid is isosceles, we know that BC=AD=7 cm BC=AD=7~cm.

The given height is h=5 cm h=5~cm .

The perimeter of the trapezoid is P=34 cm P=34~cm .

To find the sum of the two bases AB+CD AB + CD we subtract the known sides BC=AD=7 BC=AD=7 from the perimeter P P

PBCAD=AB+DC P-BC-AD=AB+DC

3477=AB+DC=20cm 34-7-7=AB+DC=20\operatorname{cm}

Now we will use that value to find the area of the trapezoid

Aˊrea=5(20)2=50 Á\text{rea}=\frac{5(20)}{2}=50

Answer:

50 cm2 50~cm²


Exercise 4

Given:

Trapezoid DECB DECB is part of triangle ABC \triangle ABC which has the following values:

AB=6 cm AB=6~cm

AC=10 cm AC=10~cm

Also, we know that DE DE intersects AB AB and AC AC respectively.

Task:

Calculate the area of trapezoid DECB DECB

Calculate the area of the new DECB trapezoid

Solution:

To find the area of the trapezoid we must first find the values of the sides BC BC and DE DE .

We will find the values usig the Pythagorean Theorem.

AB=6 cm AB=6~cm

AC=10 cm AC=10~cm

Because DE DE divides the side AB AB into two equal segments we know that AD=DB=3 cm AD=DB=3~cm

Because DE DE divides AC AC into two equal segments we know that AE=EC=5 cm AE=EC=5~cm.

According to the Pythagorean theorem we then will have:

AD2+DE2=AE2 AD^2+DE^2=AE^2

AB2+BC2=AC2 AB^2+BC^2=AC^2

By substituting the data:

32+DE2=52 3^2+DE^2=5^2

DE2=16 DE^2=16

DE=4 DE^=4

62+BC2=102 6^2+BC^2=10^2

BC2=64 BC^2=64

BC=8 BC=8

Now we can find the area of the trapezoid:

A=(DE+BC)×DB2=(4+8)×32=18 A=\frac{(DE+BC)\times DB}{2}=\frac{(4+8)\times3}{2}=18

Answer:

18cm2 18\operatorname{cm}^2


Test your knowledge

Exercise 5

Given the trapezoid ABCD ABCD

AB=AD AB=AD

The side DC DC is twice as large as the side AB AB .

The value of the area of the trapezoid is three times greater than the side AB AB .

Task:

Find the value of the side AB AB

A10 - trapezoid ABCD

Solution:

To find the side AB AB we will put the data into the formula.

X X will be the unknown variable in the equation

AB=X AB=X

h=AD=X h=AD=X since we know that AB=AD AB = AD (height)

DC=2X DC=2X since the side DC DC is 2 2 times larger than AB AB

Area=3X A\text{rea}=3X since that the trapezoidal area is 3 3 times bigger than the side AB AB

We can now substitute the values into the formula.

Area=(X+2X)×X2=3X1= A\text{rea}=\frac{(X+2X)\times X}{2}=\frac{3X}{1}=

We eliminate the fractions by multiplying the whole equation by two:

6X=X(X+2) 6X=X(X+2)

We divide both sides of the equation by X X :

6=X+2X 6=X+2X

6=3X 6=3X

X=2 X=2

Therefore, side AB=2 AB=2


Questions and answers about trapezoids

What is the formula to find the area of a trapezoid?

(Base1+Base2)×Height2=Area of trapezoid \frac{\left(Base1+Base2\right)\times Height}{2}=Area~of~trapezoid


How do you calculate the perimeter of a trapezoid?

The lengths of the four sides are added together.


What is a trapezoid?

It is a quadrilateral that has a pair of parallel opposite sides.


What is the main characteristic of a trapezoid?

It has two parallel sides.


Do you know what the answer is?

How many exercises do I need to solve before finding the area of trapezoids becomes easy?

The formula for finding the area of a trapezoid doesn't have to be complicated.
All you need to do is review until you can remember the formula, and then you will be able to input the given values from the exercises with ease.


Also, remember that it is important to follow the order of operations in their proper order: PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition and then Subtraction).

Sometimes, you may have to solve for areas of trapezoids with missing values.

However, if you are familiar with the properties of the different types of trapezoids (like isosceles and rectangular trapezoids), you will be able to find the missing values and solve for the area.

How many exercises should I practice?

Since every student learns at a different pace, the answer to this question will be individual for everyone.
The important thing is to understand your current level, and know when you need to practice more and when you're ready to move on.
Typically, we recommend starting by solving ten basic and medium level exercises in order to memorize the basic formula.


How can I prepare for pop quizzes?

The answer is quite simple.
Many students are deathly afraid of pop quizzes, but in reality, they are an opportunity to practice and show your knowledge.
The key to doing well is to study consistenly throughout the year, not just before the exams.

  • Knowing that there will be a test will usually motivate you to stay up to date on your homework.
  • Avoid falling behind on your studies, and stay on top of the latest lectures.
  • Exams questions usually start by testing your knowledge on one topic at a time. For example: calculate the area of a trapezoid.
  • Grades are based on a yearly average, so it is in your best interest to get the best possible score on each test.

As long as you pay attention in class and do your homework, you won't have to be afraid of exams.


Check your understanding

How do I know if I am falling behind?

Are there topics that you don't understand or can't remember learning? Some topics might seem easy to you, while some may seem more difficult - that's completely normal!

Important: don't over-procrastinate learning the course material. The pace in math courses can be fast and you don't want to get left behind.
Many new topics are built on what topics you've already learned. Therefore, if you haven't taken the time to make sure you know an old topic well, you might find it hard to understand the new topic.
How do you know if you are falling behind?

  • You find it difficult to stay concentrated in class because you don't understand the teacher.
  • You have difficulty doing your homework.
  • You have started getting lower scores on your exams.

What can I do to catch up?

  • You can ask a classmate to help explain what you don't understand.
  • Speak up and ask your math teacher to help you.
  • You can find a private tutor to help you strengthen the ares that you are struggling with.

Do you think you will be able to solve it?

Studying math with a private tutor

There are students who find it difficult to keep up with the pace in class.
It is important to understand that learning quickly is not necessarily related to the student's ability to understand, or even to passing exams and getting good scores.
Sometimes teachers have to teach material quickly in order to cover all the topics in their curriculum. When this happens, there are often students who do not manage to catch all the the different explanations and formulas, and little by little they fall behind.

With a private math teacher you will not only be able to improve in the areas you might be struggling with, but you will also learn to how to learn effectively.
A private teacher can help you pass your high school exams, as well as prepare you for higher education.
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Test your knowledge

examples with solutions for trapezoid area

Exercise #1

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

2.52.52.5444h=6h=6h=6AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, let's remind ourselves of the formula for the area of a trapezoid:

A=(Base + Base) h2 A=\frac{\left(Base\text{ }+\text{ Base}\right)\text{ h}}{2}

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 = 
19.5

Answer

1912 19\frac{1}{2}

Exercise #2

Given the trapezoid:

999121212555AAABBBCCCDDDEEE

What is the area?

Video Solution

Step-by-Step Solution

Formula for the area of a trapezoid:

(base+base)2×altura \frac{(base+base)}{2}\times altura

We substitute the data into the formula and solve:

9+122×5=212×5=1052=52.5 \frac{9+12}{2}\times5=\frac{21}{2}\times5=\frac{105}{2}=52.5

Answer

52.5

Exercise #3

Calculate the area of the trapezoid.

555141414666

Video Solution

Step-by-Step Solution

We use the formula (base+base) multiplied by the height and divided by 2.

Note that we are only provided with one base and it is not possible to determine the size of the other base.

Therefore, the area cannot be calculated.

Answer

Cannot be calculated.

Exercise #4

What is the area of the trapezoid in the figure?

777151515222AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

We use the formula to calculate the area of a trapezoid: (base+base) multiplied by the height divided by 2:

(AB+DC)×BE2 \frac{(AB+DC)\times BE}{2}

(7+15)×22=22×22=442=22 \frac{(7+15)\times2}{2}=\frac{22\times2}{2}=\frac{44}{2}=22

Answer

22 22 cm².

Exercise #5

What is the area of the trapezoid in the figure?

222999777AAABBBCCCDDD

Video Solution

Step-by-Step Solution

We use the formula: (base + base) multiplied by height divided by 2:

S=(AB+DC)×h2 S=\frac{(AB+DC)\times h}{2}

Keep in mind that AD is the height of a trapezoid:

We replace the existing data in the formula:

S=(2+9)×72 S=\frac{(2+9)\times7}{2}

S=11×72=772=38.5 S=\frac{11\times7}{2}=\frac{77}{2}=38.5

Answer

38.5 38.5 cm².

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