# Area of Isosceles Triangles

🏆Practice area of a triangle

## Formula to calculate the area of an isosceles triangle

$\frac{Height~of~the~base~\times ~Base}{2}=A$

## Test yourself on area of a triangle!

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

## Area of Isosceles Triangles

Calculating the area of an isosceles triangle is very simple, easy, and even identical to the calculation we do to find out the area of other types of triangles. Therefore, if you happen to get a question about calculating the area of isosceles triangles on the exam, I assure you that a small smile will appear on your face.

### How is the area of an isosceles triangle calculated?

We will multiply the base by the height and divide by two.

Remember!

The main property of the isosceles triangle is that the median of the base, the bisector, and the height are the same, that is, they coincide. Therefore, even if the question only names the median of the base or the bisector, you can immediately deduce that it is also the height of the triangle and use it to calculate its area.

Observe the theorem holds true only with the height, the median of the base, and the bisector!

You didn't think we were going to send you off without any exercises on the topic, did you? Time to practice!

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Here you have an isosceles triangle $ABC$

Given that:
$ab=ac$
$AD$ -

Height
$AD = 4$
$CB=6$

What is the area of the triangle?

Solution: We will proceed according to the formula - the height $AD = 4$
multiply by the base $CB = 6$
and divide the received product by $2$
We will obtain:
$\frac{4\times6}{2}=12$
The area of the triangle $ABC$ is $12$ cm2.

### Now let's move on to an exercise that aims to be a bit more sophisticated:

You have the isosceles triangle $FDC$

Given that:
$FC=FD$
$CG= 4$
$FG = 5$ The median of the base

Calculate the area

Solution: Let's remember that, in an isosceles triangle, the median of the base is also the height, therefore, we can use it in the formula for the area of the isosceles triangle. Let's note: Height $FG=5$
Now let's see that we have only half of the base $CG =4$ .
Since $FG$ is given as the median, we can deduce that also $GB=4$ and consequently, the entire side of the base $CD=8$
Now let's put it in the formula:
$\frac{4\times 8}{2}=16$
The area of the triangle $FDC$ is $16$ cm2 .

Do you know what the answer is?

### Bonus exercise (tip) for advanced level

Formula to calculate the area of an isosceles triangle that is also a right triangle:

If you come across calculating the area of an isosceles triangle whose height has not been given, but you know it is a right triangle, it is useful to know the following trick:

Let's see how it is done by applying it in an exercise: Before you, you have an isosceles right triangle $ABC$
Given that $AB=AC$
angle $ABC = 90$
$AB=4$

Calculate the area of the triangle

Solution: Let's not be scared of not having data about the height and proceed according to the formula: the triangle is isosceles, therefore $AB=AC=3$.

These are the two legs of the triangle - they form a right angle. Consequently, we will obtain:
$\frac{4\times4}{2}=8$
The area of the triangle is $8$ cm2 .

## Examples and exercises with solutions for the area of an isosceles triangle

### Exercise #1

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:

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

We insert the existing data as shown below:

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

10

### Exercise #2

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 #3

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:

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

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

36 cm²

### Exercise #4

Calculate the area of the following triangle:

### Step-by-Step Solution

The formula for the area of a triangle is

$A = \frac{h\cdot base}{2}$

Let's insert the available data into the formula:

(7*6)/2 =

42/2 =

21

21

### Exercise #5

Calculate the area of the right triangle below:

### Step-by-Step Solution

Due to the fact that AB is perpendicular to BC and forms a 90-degree angle,

it can be argued that AB is the height of the triangle.

Hence we can calculate the area as follows:

$\frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24$