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.

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 parallelsides.

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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 $A$ and ending at vertex $B$ is $6~cm$.

The length of the base starting at vertex $D$ and ending at vertex $C$ is $10~cm$.

The height (represented by the letter $h$) is $4~cm$.

The sum of the bases is

$16$$(6+10)$.

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

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

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

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

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 $A$ and ending at the vertex $B$ is $6~cm$.

The length of the base that starts at vertex $D$ and ends at vertex $C$ is $9~cm$.

The height between the bases (represented by the letter h) is $4~cm$.

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

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

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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.

Below we are given a trapezoid with the following values:

What is its height?

Solution

Formula for the area of a trapezoid:

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

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

$\frac{9+6}{2}\times h=30$

Now to reduce and solve:

$\frac{15}{2}\times h=30$

$7\frac{1}{2}\times h=30$

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

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

$h=4$

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

Exercise 2

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

$DC=14~cm$

$AD=5~cm$

$KB=4~cm$

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

$DC = 14~cm$

$AD = 5~cm$

$KB = 4~cm$

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.

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:

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

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

The height is $CB$.

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

Therefore $CB = 5$.

Now we can calculate

$4\times 5=20$

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

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

Question 1

What is the area of the trapezoid in the diagram below?

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!

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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.

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