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.

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!

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.

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 .

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 .