The Area of An Equilateral Triangle

The way to find the area of an equilateral triangle is the same as finding the area of other triangles.



$$\text{The area of a triangle :}\frac{\text{Base x Base Height}}{2}$$

According to the above formula, we need to know the base and base height.
However is it possible to calculate one or the other if we don’t know its value?
For example, generally the base (of an equilateral triangle) is known but the
base height is not known. The main problem is; can we know the height by
using the base value?


First of all, let us look at the height of isosceles and equilateral triangles. Let us have a
triangle where the base is equal but the arms can be flexible .





When both sides are equal then the height is exactly joint together in the middle point.



The yellow and blue length are equal, the green height has split the equal sides down
the middle. Now our triangle is an isosceles triangle, if the black side is equal to the
other sides it becomes an equilateral triangle.


We can see that with equilateral triangles that the middle line splits the base into equal
two parts. Now let’s look at the angles;


The height splits the equal two parts from 60° into 30°.
We can understand this in two ways.


1st Method :


Let’s subtract the other two angles 
\( \displaystyle 90°+60°=150° \) from \( \displaystyle 1800° \)
the total value of the angles  of a triangle. \( \displaystyle 180°-150°=30° \)
It can be found in the same way on the other
angle.


2nd Method

Imagine opening the wings slowly.

As the base value increases, the
top angle increases also.


If I increase the length here, the angle above increases also.

So, as the bottom lengths are the same, the angles must also be the same. 
I can divide 60° into pairs of 30° and 30° .

We have learnt;
  • In isosceles triangles or equilateral triangles, the base height divides the base length into two equal parts.
  • The height also divides the angles into equal values.
So, let us collect the info we have from equilateral triangles;


Our aim is to find h, the height…

To find the height of a right-angled triangle , we can use the Pythagorean theorem.


Let us remember the Pythagorean relation, in a right-angled triangle, the square of the
right-angled sides total equal the longer side..


Let us apply the Pythagorean relation to an equilateral triangle,

$$h^2+\left(\frac{a}{2}\right)^2=a^2$$
$$h^2+\frac{a^2}{4}=a^2$$
$$h^2=a^2-\frac{a^2}{4}$$
$$h^2=\frac{4a^2}{4}-\frac{a^2}{4}$$
$$h^2=\frac{3a^2}{4}$$
$$\sqrt{h^2}=\sqrt{\frac{3a^2}{4}}$$
$$h=\frac{a\sqrt{3}}{2}$$

Lets go back to our formula ;
$$\text{The area of a triangle :}\frac{\text{Base x Base Height}}{2}$$

$$\frac{a.\frac{a\sqrt{3}}{2}}{2}=\frac{\frac{a^2\sqrt{3}}{2}}{2}=\frac{a^2\sqrt{3}}{2}.\frac{1}{2}=\frac{a^2\sqrt{3}}{4}$$

So the formula of equilateral triangle ;

" a " represents one side of an equilateral triangle ;
$$\frac{a^2\sqrt{3}}{4}$$


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