Area of Equilateral Triangle, Formula, Examples & Derivation

The Area of the Equilateral Triangle is quite an important topic for solving mathematical problems and formulas, especially in the calculation of the space occupied between the sides of the equilateral triangle in a 2D plane. In an equilateral triangle, the median, angle bisector, and altitude are the same for all sides and are the triangle's lines of symmetry. Finding the area of an equilateral triangle is quite an easy calculation. 

We all know what a triangle is; it is categorised into three types: scalene triangles, equilateral triangles, and isosceles triangles.

• A scalene triangle has no equal sides or angles.
• Two sides of an isosceles triangle are equal, as are the opposing angles of equal sides.

Area of an Equilateral Triangle

The area of an equilateral triangle is the amount of space that the triangle takes up on a two-dimensional plane. We all know the triangle as the most elementary type of a regular polygon. The word "triangle" comes from the fact that it has three angles, formed by the joining of three line segments end to end triangle. A closed geometric object having three sides, three vertices and three angles. And the sum of the three angles of a triangle is 180 degrees. The length of a triangle’s sides determines its type.

This triangle has three equal sides and is hence known as an equilateral triangle. As a result,

each equilateral triangle angle is 60 degrees.

Area of Equilateral Triangle Formula

The formula for the area of an equilateral triangle (A) is given below,

A = (√3/4)a²

Where a is the length of the sides of an equilateral triangle.

To find the area of the Equilateral Triangle, you must know the measure of the side length of the equilateral triangle.

Apply the formula to calculate the equilateral triangle's area given as A = (√3/4)a², where a is the measure of the side length of the equilateral triangle.

What is an equilateral triangle?

Yes, in simple words, an equilateral triangle is a triangle with three equal sides and three interior angles that each measure 60 degrees. 

Other equilateral triangle properties are the following:

• The perimeter of an equilateral triangle is 3s. Where 's' is the equilateral triangle's side.

• The ortho-centre and centroid of the triangle are at the same point.

• In an equilateral triangle: the median, angle bisector, and perpendicular are all identical.

Derivation of Area of the Equilateral Triangle

Let’s derive the Area of the equilateral triangle.

We are all familiar with the primary area formula for triangles, that is,

Area of a triangle = 1⁄2 × height of the triangle × base of the triangle .... (i)

Here, height = h and base = a

By using this formula, we can easily calculate the area of an equilateral triangle.

Now let us consider an equilateral triangle with sides (a) and height (h).

By using Pythagoras' Theorem in the triangle, we get

H² = P² + B² .... (ii)

By putting the values in equation (ii) according to the diagram, we get

a² = h² + (a / 2)²

h² = a² - (a² / 4)

h² = (3a²  / 4)

h = ½ (√3a)

Now, put the value of "h" in equation (i)

Area of Triangle = 1⁄2 × height of the triangle × base of the triangle

S = ½ x ½ (√3a) x a

Area of Equilateral Triangle = ¼ (√3a²)

The perimeter of an Equilateral Triangle

Because all sides of an equilateral triangle are equal, the perimeter of an equilateral triangle equals the sum of all sides or three times of a side.

The perimeter of an Equilateral Triangle = 3a,

where a is the side

we can also remember:

• Semi-perimeter of an Equilateral Triangle = 3a/2

1. • Height of an Equilateral Triangle = √3a/2