Area of a right-angled trapezoid

Before we begin, let's recall some properties of a right-angled trapezoid:
Properties of a right-angled trapezoid

• In a right-angled trapezoid, there are 2 angles equal to 90 degrees each.
• The leg connecting the 2 right angles is also the height of the trapezoid!
• In a right-angled trapezoid - the total sum of angles is 360 degrees, where 2 angles are equal to 90 degrees each and the other 2 angles sum up to 180.

Let's see it in an illustration:

To calculate the area of a right-angled trapezoid, we will use the following formula:

And in words: the area of a right-angled trapezoid equals the sum of the bases multiplied by the height divided by 2.

Exercise:
Given the following trapezoid, calculate its area.

Given:
angle $A = 90$
angle $D = 90$

$AB= 2$
$DC= 4$
$AD= 3$

Solution:
We are given that there are 2 right angles in the trapezoid, therefore it is a right-angled trapezoid.
To calculate the area of the trapezoid, we need to add the two bases, multiply by the height and divide the result by 2.
We know that in a right-angled trapezoid, the height is also the side connecting the two right angles, meaning $AD = 3$.
Therefore:
We'll add the given bases $AD + CD$ and multiply by the height $AD$ and divide this by $2$. We'll get:
$\frac{(2+4) \cdot3} {2} =$

$\frac{18}{2}=9$
The area of the trapezoid is $9$ cm².

Additional Exercise

Here is the following right-angled trapezoid:

Given that:
Angle $A = 90$
Angle $B = 100$
Angle $C = 80$
$AD= 2$
$AB = 5$
$DC = AB+1$

What is the area of the trapezoid?

Solution:
First, we need to look at all the given information and understand what type of trapezoid we are dealing with.
We are given one angle equal to $90$ degrees and $2$ other angles that together equal $180$ degrees.
We know that the sum of angles in a trapezoid equals $360$ degrees, therefore angle $D$ must equal $90$ degrees.
Now we know that in this trapezoid there are $2$ angles that equal $90$ degrees each, therefore it is a right-angled trapezoid.
To calculate the area of a right-angled trapezoid, we need to know the lengths of the bases and the height.
The height in a right-angled trapezoid is also the side connecting the two right angles - meaning side $AD= 2$
The two bases are: $AB$ and $DC$
According to the given information: $AB = 5$ and $DC= AB+1$
Therefore
$DC = 6$
Let's calculate using the right-angled trapezoid area formula and we'll get that:

$\frac{(6+5) \cdot2} {2} = 11$

The area of the trapezoid is $11$ cm².

Additional Exercise:

Given that:
The area of the trapezoid is $25$ cm²
Angle $D= 20$
Angle $A = 90$
Angle $B = 90$
Known that the sum of bases is $25$.
Find the length of side $AB$ and the size of side $C$.

Solution:
We can immediately identify that this is a right-angled trapezoid since it has $2$ angles equal to $90$ each.
We are given the area of the trapezoid and we need to find the height - $AB$
If we recall the formula for finding the area of a right-angled trapezoid and substitute the sum of the bases and the given area of the trapezoid, we get that:

$\frac{(25) \cdot AB} {2} =25$
We can clearly see that $AB$ must be $2$ in order to get a true statement, therefore the height of the trapezoid $AB$ equals $2$.

The size of side $C$ needs to be completed to $180$.
It is known that angle $D$ equals $20$ and therefore $C$ equals $160$.