Equilateral triangle is a triangle in which all the three sides are of equal length. Equilateral triangles are equiangular, that is the interior angles are congruent to each other and each angle will be equal to 60$^{0}$ and doesn't matter how big or small the triangle is.

For example, if only two angles are known to be each of 60$^{0}$ then the third angle has to be 60$^{0}$ as triangle is an equilateral triangle.

## Properties of an Equilateral Triangle

Equilateral triangles are regular polygons and are known as regular triangles. Property of regular triangles also apply to equilateral triangles. Given below are the important properties of an equilateral triangle.
• All three angles of an equilateral triangle are always 60$^{0}$.
• Equilateral triangles are bilaterally symmetrical in three different ways.
• Interior angles of a triangle add up to 180$^{0}$ and exterior angles of a triangle add up to 360$^{0}$.
• The formula to find the area of an equilateral triangle is given by $\frac{\sqrt{3}}{4}$ S$^{2}$ where 'S' is the length of any one side.
• In equilateral triangle the radius of the incircle is exactly half the radius of the circumcircle.
• Equilateral triangle have three lines of symmetry from each vertex to the midpoint of the opposite side. Lines are known to be as medians, perpendicular bisectors, altitudes and angle bisectors of the triangle.
• The point where in the three lines intersect is the centroid, incenter, circumcenter and the orthocenter of the triangle.
• All equilateral triangles are also isoceles triangles and if you stack six equilateral triangles together you will get a hexagon.
• The radius of the circumscribed circle is R = $\frac{\sqrt{3}}{3}$a and the radius of the inscribed circle is $\frac{\sqrt{3}}{6}$a.

## Area of an Equilateral Triangle

Area of an equilateral triangle is found by using pythagoraen theorem.

Here BC = BD + DC and BD = DC = $\frac{S}{2}$

Apply Pythagorean Theorem in $\triangle$ ABD.

$(AB)^2 = (BD)^2 + (AD)^2$

We have AD = h, AB = s, BD= $\frac{s}{2}$

$S^{2}$ = $(\frac{S}{2})^{2}$ + $h^{2}$

$S^{2}$ = $\frac{S^{2}}{4}$ + $h ^{2}$

$S^{2}$ - $\frac{S^{2}}{4}$ = $h^{2}$

$\frac{3S^{2}}{4}$
=  $h^{2}$

h = $\frac{\sqrt{3}}{2}$ S
We know that, the area of a triangle  = $\frac{(b \times h)}{2}$    Area = $\frac{(S \times \sqrt{3} \times S)} {4}$      (Substitute the values of b and h)

= $\frac{S^{2} \times \sqrt{3}}{4}$

Area of an equilateral triangle =  $\frac{S^{2} \times \sqrt{3}}{4}$ Sq. units

## Example of Equilateral Triangle

### Solved Examples

Question 1: Find the area of an equilateral triangle with a side of 2.5 cm.
Solution:

Given S = 2.5 cm
The formula to find the area of an equilateral triangle is

A = $\frac{\sqrt{3}}{4}$ S$^{2}$

Plug in the given value and solve.

A = $\frac{\sqrt{3}}{4}$ (2.5)$^{2}$

A = 2.70

Therefore the area of an equilateral triangle is 2.70 cm$^{2}$

Question 2: Find the area of an equilateral triangle with a side of 140 cm.
Solution:

Given S = 140 cm
The formula to find the area of an equilateral triangle is

A = $\frac{\sqrt{3}}{4}$ S$^{2}$

Plug in the given value and solve.

A = $\frac{\sqrt{3}}{4}$ (140) $^{2}$

A = 8428

Therefore the area of an equilateral triangle is 8428 cm$^{2}$

Question 3: If the area of an equilateral triangle is 390 cm$^{2}$ find the side in it.
Solution:

Given A = 390 cm$^{2}$
The formula to find the area of an equilateral triangle is

A = $\frac{\sqrt{3}}{4}$ S$^{2}$

Plug in the given value and solve.
390  = $\frac{\sqrt{3}}{4}$ S$^{2}$

$\frac{1560}{1.732}$ = S$^{2}$

or S = 30.01

Therefore the side of an equilateral triangle is 30.01 cm