# Area of a Scalene Triangle

🏆Practice triangle area

## Area of a scalene triangle

Formula to calculate the area of a scalene triangle:

## Test yourself on triangle area!

What is the area of the given triangle?

## Area of the scalene triangle

It's very simple to calculate the area of a scalene triangle if we remember the formula and strictly follow the steps. Don't worry, we're here to teach you exactly what to pay attention to—we won't leave you adrift!
First of all, let's look at the formula you need to remember in order to calculate the area of the scalene triangle:

Multiply the height by the base (the side corresponding to that height) and divide by $2$.

Pay attention:

Make sure to place in the formula the corresponding height and side. That is, if a certain height and a side that does not form a right angle of $90^o$ degrees with the used height is placed in the formula, it will be wrong.

### Let's see it in an exercise

Given the triangle $ABC$
Given that:
$DB=6$ Height
$AC = 7$
What is the area of the triangle?

Solution:
We will see that the given side $AC$ actually forms, with the height, an angle of $90^o$ degrees.
After verifying the data, we will go to the formula and place there:
$\frac{6\times7}{2}=21$

The area of the triangle $ABC$ is $21\operatorname{cm}^2$

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today

### Now we will calculate the area of a right triangle

Given the right triangle $EFG$
Given that:
angle $EFG = 90$
$EF=5$
$FG=6$
Calculate the area of the triangle.

Solution:
Let's remember that the key to calculating the area of any triangle is to multiply the height
by the corresponding side and then divide that product by$2$
In a right triangle, we actually already have the height!
We don't need to calculate another height and, in fact, we can afford to use the given height along with the side that forms the $90^o$ degree angle.

In our exercise: The side is $EF$ or $FG$

What conclusion do we reach?
The conclusion is that the formula to calculate the area of a right triangle is the product of the two legs divided by $2$Let's put it in the formula and we will get:

$\frac{6\times5}{2}=15$
The area of the triangle $EFG$ is $15cm^2$

### Now let's move on to calculating the area of an obtuse triangle

Calculating the area of an obtuse triangle is a bit more complicated, but I assure you that once you understand the basic principle, you will be able to calculate the area of an obtuse triangle even in your sleep...
In certain cases, in an obtuse triangle, we will be given a height that is outside the triangle.
As in the following illustration:

In this illustration, the height $AG$ has been drawn outside of the triangle. In reality, if we were to extend the side $CB$ (marked in green), it would form a right angle with the height.
How is the area of an obtuse triangle calculated?

Remember the following guidelines and you will do well:

• In calculating the area of the obtuse triangle, we refer to the actual side length of the triangle and not to its dotted extension.
• In calculating the area of the obtuse triangle, we refer to the given height (even if it is outside the triangle) and look for the corresponding side, which together with it forms a $90^o$ degree angle when extended outside the triangle.

Now let's solve an exercise so you can understand it more easily:

Given the triangle $\triangle ABC$
Given that:
$BD= 2$ Height of the triangle
$AD= 5$
$CD= 12$

What is the area of the triangle?

Solution:
We observe that the length of the side $DB = 2$
and the corresponding side that forms with it a $90^o$ degree angle (the dotted part outside the triangle) is $CA$
If we go back to the first point we needed to remember - we will understand that, to calculate the area, we must only take into account the length of $AC$ without its dotted extension.
Therefore, we will see it as $12-5=7$
$AC=7$
And now we can safely place the data, according to the basic formula:
$\frac{7\times2}{2}=7$
The area of the triangle $ABC$ is $7cm^2$

## Examples and exercises with solutions for calculating the area of a scalene triangle

### Exercise #1

What is the area of the given triangle?

### Step-by-Step Solution

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

The height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

$\frac{6\times5}{2}=\frac{30}{2}=15$

15

### Exercise #2

Calculate the area of the triangle below, if possible.

### Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Cannot be calculated

### Exercise #3

Calculate the area of the following triangle:

### Step-by-Step Solution

The formula for the area of a triangle is

Let's plug in the data we have into the formula:

(7*6)/2 =

42/2 =

21

21

### Exercise #4

Calculate the area of the following triangle:

### Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

$\frac{BC\times AE}{2}$

Now we replace the existing data:

$\frac{4\times5}{2}=\frac{20}{2}=10$

10

### Exercise #5

Calculate the area of the triangle ABC using the data in the figure.

### Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

(the side * the height that descends to the side) /2

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

$\frac{CB\times AD}{2}$

$\frac{8\times9}{2}=\frac{72}{2}=36$