# Area of Isosceles Triangles

🏆Practice triangle area

## Test yourself on triangle area!

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

## 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

### examples.example_title

Find the area of the triangle (note that this is not always possible)

### examples.explanation_title

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.

### examples.solution_title

Cannot be calculated

### examples.example_title

Which of the following triangles have the same area?

### examples.explanation_title

We calculate the area of triangle ABC:

$\frac{12\times5}{2}=\frac{60}{2}=30$

We calculate the area of triangle EFG:

$\frac{6\times10}{2}=\frac{60}{2}=30$

We calculate the area of triangle JIK:

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

It can be seen that after the calculation, the areas of the similar triangles are ABC and EFG

EFG, ABC

### examples.example_title

Given the triangle PRS

The length of side SR is 4 cm

The area of the triangle PSR is 30 cm²

Calculate the height PQ

### examples.explanation_title

We use the formula to calculate the area of the triangle.

Pay attention: in the obtuse triangle, its height is located outside the triangle!

$\frac{Lado\cdot\text{Altura}}{2}=Área~del~triangulo$

Double the equation by a common denominator.

$\frac{4\cdot PQ}{2}=30$

$\cdot2$

Divide the equation by the coefficient of $PQ$.

$4PQ=60$ / $:4$

$PQ=15$

15 cm

### examples.example_title

Find X using the data from the figure:

### examples.explanation_title

The formula to calculate the area of a triangle is:

(side * height descending from the side) /2

We place the data we have into the formula to find X:

$20=\frac{AB\times AC}{2}$

$20=\frac{x\times5}{2}$

Multiply by 2 to get rid of the fraction:

$5x=40$

Divide both sections by 5:

$\frac{5x}{5}=\frac{40}{5}$

$x=8$

8

### examples.example_title

The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

It is known that the ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC

### examples.explanation_title

To calculate the ratio between the sides we will use the existing figure:

$\frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}$

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

$A_{ADE}=\frac{h\times DE}{2}$

We know that the area of the parallelogram is equal to:

$A_{ABCD}=h\times EC$

We replace the data in the formula given by the ratio between the areas:

$\frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}$

We solve by cross multiplying and obtain the formula:

$h\times EC=3(\frac{1}{2}h\times DE)$

We open the parentheses accordingly

$h\times EC=1.5h\times DE$

We divide both sides by h

$EC=\frac{1.5h\times DE}{h}$

We simplify to h

$EC=1.5DE$

Therefore, the ratio between$\frac{EC}{DE}=\frac{1}{1.5}$

### examples.solution_title

$1:1.5$