# Area of Equilateral Triangles

🏆Practice triangle area

Formula to calculate the area of an equilateral triangle:

## Test yourself on triangle area!

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

## Area of an Equilateral Triangle

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:
$\frac{6\times3}{2}=9$

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

Simple and easy, right?

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### Now let's move on to a more complicated exercise.

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:

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

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

## Examples and exercises with solutions for calculating the area of an equilateral 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

$1:1.5$

Do you know what the answer is?
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