Area of Equilateral Triangles

Calculating the area of an equilateral triangle is quite simple, you can't get too confused with it, not even a little.
All you need to remember is the formula we will present to you below and apply it to equilateral triangles:

Remember!
In equilateral triangles, the height is also the median and the bisector.
Therefore, if the question only gives the length of the median or the bisector, you can immediately deduce that it is the height you need to place in the formula.
And on top of that, since the triangle is equilateral, you can immediately find the length of the edge (or side) corresponding. Simply compare it with the given edge since they are all equivalent.

Let's practice so we can understand even better how to calculate the area of an equilateral triangle:

Given the triangle $\triangle ABC$

Given that:
$ABC$ Equilateral triangle
$AD=3$ Height
$CB= 6$

What is the area of the triangle?

Solution:
At first glance, we see that we have a height equivalent to $3$ and a side equivalent to $5$.

Let's put it in the formula and we will obtain:
$\frac{6\times3}{2}=9$

Answer:

The area of the triangle $ABC$ is $9$ cm².

Simple and easy, right?

that covers various hypothetical situations that could confuse you on the exam:

Given the equilateral triangle $\triangle ABC$

Given:
$AC = 6$
$DB=AD$
$CD=7$

What is the area of the triangle $\triangle ABC$?

Solution:

We know that to calculate the area of the triangle, we need to have the length of the height and the corresponding side with which it forms $90^o$ degrees.

In this exercise, it is not explicitly stated that $CD$ is the height of the triangle, but we know that: $AD =DB$ that is, $CD$ is the median - it crosses the side it touches, dividing it into two equal parts.
Since it is an equilateral triangle, the median is also the height of the triangle, and therefore, we can use it in the formula for calculating the area.
Additional note: If instead of the data that $CD$ is the median, they had given that it is the bisector $ABC$, we could also have deduced that it is the height, since in an equilateral triangle, the median, the height, and the bisector coincide.

Therefore, we will note $CD=7$ as the height of the triangle.

Now we must find the length of the side $AB$
Since it is an equilateral triangle, all sides are equal, so we immediately deduce that $AB=AC = 6$
Now let's put it in the formula and we will get:

$\frac{6\times7}{2}=21$

Answer:
The area of the triangle $ABC$ is $21$ cm².