# Perimeter of a trapezoid

🏆Practice perimeter of a trapezoid

The trapezoid is a quadrilateral defined as having 2 parallel opposite sides. The calculation of the perimeter of the trapezoid is solved using a very simple formula that we will see below: all sides are added together. This type of questions can appear in tests of the first and second level in the first years of high school and also in final exams of level 3, 4 and 5 for the graduation of the secondary cycle.

## Test yourself on perimeter of a trapezoid!

Calculate the perimeter of the trapezoid according to the following data:

## Exercises

### Exercise 1 (Examples of calculating the perimeter of a trapezoid)

Suppose we are presented with the following data about the sides of a trapezoid in a question:

Homework:

Let's see then, how do you calculate the perimeter of a trapezoid?

Solution:

$A=5$
$B=3$
$C=4$
$D=6$

in such a case the calculation would be: $5+3+4+6=18$. And here it is: the perimeter of the trapezoid is $18$

### Exercise 2

$A=2$
$B=3$
$C=4$
$D=4$

Solution:
In such a case the calculation would be: $2+3+4+4=13$. Here, the perimeter of the trapezoid is $13$

Answer: $13$

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### Exercise 3

The sides of the trapezoid as they appear in the question:

What is the perimeter of the trapezoid?

Solution:

$A=9$
$B=8$
$C=6$
$D=4$

We calculate all the sides: $9+8+6+4=27$. That is, the perimeter of the trapezoid is $27$.

$27$

### Exercise 4

Given that the perimeter of the trapezoid is $30$.

$A=7$
$B=?$
$C=5$
$D=10$

What is the length of the side $B$?

Solution:

We calculate all sides: $7+5+10=22$. Let's see, $30-22=8$. Then, the length of the side $B$ is $8$.

Pay attention: The mathematical operation is a simple addition. But, you must take into account the following:

You have to know the properties of the trapezoid to fill in the missing sides.

You have to know by heart the formula to calculate the perimeter of the trapezoid.

$B=8$

Do you know what the answer is?

### Exercise 5

Given the isosceles triangle $\triangle ABC$,

In its interior is plotted $EF$:

$AF=5 AB=17$

$AG=3 AD=8$

What is the perimeter of the trapezoid $EFBC$?

Solution:

To find the perimeter of the trapezoid, it is necessary to add up all its sides.

We will focus on finding the bases.

To find $GF$, we will use the Pythagorean theorem: $A^2+B^2=C^2$ in the triangle $\triangle AFG$.

We replace:

$3^2+GF^2=5^2$

We isolate GF and solve:

$9+GF^2=25$

$GF^2=25-9=16$

$GF=4$

We operate the same process with the side $DB$ in the triangle $\triangle ABD$:

$8^2+DB^2=17^2$

$64+DB^2=289$

$DB^2=289-64=225$

$DB=15$

We start by finding the side $FB$:

$FB=AB-AF=17-5=12$

Now, we reveal $EF$ and $CB$:

$GF=GE=4$

$DB=DC=15$

This is because in an isosceles triangle, the height divides the base into two equal parts. Therefore:

$EF=GF\times2=4\times2=8$

$CB=DB\times2=15\times2=30$

What remains is to calculate:

$30+8+12\times2=30+8+24=62$

$62$

### Exercise 6

Given the trapezoid in the figure

What is its perimeter?

Solution:

To calculate the perimeter of the trapezoid we will add up all its sides:

$10+12+7+7=36$

$36$

### Exercise 7

Given the trapezoid in the drawing

Given that the perimeter is equal to $26$.

What is the value of $X$?

Solution:

The perimeter of the trapezoid is equal to the sum of its sides.

To answer the question we will put the sum of the sides in an equation calculating the perimeter of the trapezoid:

$10+6+X+X+1+X=26$

We arrange the equation so that $X$ is on one side and the numbers are on the other:

$X+X+X=26-1-10-6$

$3X=9$ We divide by $3$

$:3$

$X=3$

Answer: $X=3$

### Exercise 8

Given the trapezoid:

Given: the trapezoid $ABCD$ is part of a rectangle.

Data in cm $DC=12,BK=3$

Height of the trapezoid $H=4$

Calculate the perimeter of the trapezoid.

Solution:

To find the perimeter of the trapezoid we will calculate by using the Pythagorean theorem the side $BC$.

$BC=AD$

Given that:

$KC=4$

$BK=3$

$DC=12$

$KC=4$

$AB=DC-3-3=6$

$BK²+KC²=BC²$

$3²+4²=BC²$

$9+16=BC²$

$BC=\sqrt{25}=5$

$BC=5$

Given that $BK=3$ then the segment $AB=6$

$AB=6$
$BC=5$

$CD=12$

$AD=5$

With these measures now we are going to calculate the perimeter of the trapezoid

$P=AB+BC+CD+AD$

Substituting the values:

$P=6+5+12+5=28$

$28$

Do you think you will be able to solve it?

## Review questions

What is a trapezoid?

A trapezoid is a quadrilateral that has 4 sides, two of which are bases and one of which is always larger than the other.

How to calculate the perimeter of a trapezoid?

As we know the perimeter of any geometric figure is the sum of all its sides, so in a trapezoid to calculate its perimeter just add the measures of its four sides.

What is the formula for the perimeter and area of a trapezoid?

By definition of perimeter is to add all its sides, then let it be the following trapezoid:

$P=a+b+c+d$

To calculate the area of the trapezoid, which is to calculate the surface area, we will call the side $c$ of the trapezoid as $\text{Base mayor}$ and side a as $\text{base menor}$, $h$ will be the $\text{altura}$, then the formula of the trapezoid is as follows:

$A=\frac{\left(Basemayor+basemenor\right)\times h}{2}$

$A=\frac{\left(a+c\right)h}{2}$

How to calculate the area and perimeter of a trapezoid?

To calculate perimeter and area of a trapezoid let's look at the following example

Example

Let be the following trapezoid with the following values

Assignment

Calculate perimeter and area of the trapezoid.

Solution

First we are going to calculate the perimeter, then we are going to add up all its sides.

$P=8\operatorname{cm}+10\operatorname{cm}+15\operatorname{cm}+10\text{ cm=43 cm}$

Now let's calculate the area:

We are going to add the large base plus the small base and multiply it by the height and then divide it by two.

$A=\frac{\left(a+c\right)h}{2}$

$A=\frac{\left(15\operatorname{cm}+8\operatorname{cm}\right)7\operatorname{cm}}{2}$

$A=\frac{\left(23\operatorname{cm}\right)7\operatorname{cm}}{2}$

$A=\frac{161cm^2}{2}=80.5cm^2$

$P=43\operatorname{cm}$
$A=80.50\operatorname{cm}^2$