Area of Equilateral Triangles

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Formula to calculate the area of an equilateral triangle:

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

121212888999AAABBBCCCDDD

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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:

Let's start with a classic exercise for beginners.

Given the triangle ABC \triangle ABC

Given that:
ABCABC Equilateral triangle
AD=3AD=3 Height
CB=6CB= 6

What is the area of the triangle?

Solution:
At first glance, we see that we have a height equivalent to 33 and a side equivalent to 55.

Let's put it in the formula and we will obtain:
6×32=9 \frac{6\times3}{2}=9

Answer:

The area of the triangle ABCABC is 99 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 ABC \triangle ABC

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

What is the area of the triangle ABC \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 90o 90^o degrees.

In this exercise, it is not explicitly stated that CDCD is the height of the triangle, but we know that: AD=DBAD =DB that is, CDCD 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 CDCD is the median, they had given that it is the bisector ABCABC, 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=7CD=7 as the height of the triangle.

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

6×72=21 \frac{6\times7}{2}=21

Answer:
The area of the triangle ABCABC is 2121 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)

8.58.58.5777

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?

101010121212555131313555888121212666666FFFEEEGGGCCCBBBAAAKKKJJJIIIDDDLLLHHH

examples.explanation_title

We calculate the area of triangle ABC:

12×52=602=30 \frac{12\times5}{2}=\frac{60}{2}=30

We calculate the area of triangle EFG:

6×102=602=30 \frac{6\times10}{2}=\frac{60}{2}=30

We calculate the area of triangle JIK:

6×52=302=15 \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

examples.solution_title

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

S=30S=30S=30444PPPRRRSSSQQQ

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!

LadoAltura2=Aˊrea del triangulo \frac{Lado\cdot\text{Altura}}{2}=Área~del~triangulo

Double the equation by a common denominator.

4PQ2=30 \frac{4\cdot PQ}{2}=30

2 \cdot2

Divide the equation by the coefficient of PQ PQ .

4PQ=60 4PQ=60 / :4 :4

PQ=15 PQ=15

examples.solution_title

15 cm

examples.example_title

Find X using the data from the figure:

S=20S=20S=20555XXXAAABBBCCC

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=AB×AC2 20=\frac{AB\times AC}{2}

20=x×52 20=\frac{x\times5}{2}

Multiply by 2 to get rid of the fraction:

5x=40 5x=40

Divide both sections by 5:

5x5=405 \frac{5x}{5}=\frac{40}{5}

x=8 x=8

examples.solution_title

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

AAABBBCCCDDDEEE

examples.explanation_title

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

AAEDAABCE=13 \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:

AADE=h×DE2 A_{ADE}=\frac{h\times DE}{2}

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

AABCD=h×EC A_{ABCD}=h\times EC

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

12h×DEh×EC=13 \frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}

We solve by cross multiplying and obtain the formula:

h×EC=3(12h×DE) h\times EC=3(\frac{1}{2}h\times DE)

We open the parentheses accordingly

h×EC=1.5h×DE h\times EC=1.5h\times DE

We divide both sides by h

EC=1.5h×DEh EC=\frac{1.5h\times DE}{h}

We simplify to h

EC=1.5DE EC=1.5DE

Therefore, the ratio betweenECDE=11.5 \frac{EC}{DE}=\frac{1}{1.5}

examples.solution_title

1:1.5 1:1.5

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