Area of an Equilateral Triangle: Formula, Derivation, and Examples

Calculating the area of an equilateral triangle is a fundamental mathematical skill for determining the 2D space enclosed by its sides. In this special type of triangle, the median, angle bisector, and altitude coincide for every side, serving as lines of symmetry. Fortunately, finding its area is straightforward once you know the core principles. 

We are all familiar with triangles, which are broadly categorized 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 represents the total two-dimensional space it occupies. As the most fundamental regular polygon, a triangle is a closed geometric figure defined by three line segments meeting at three vertices to form three angles, with an interior angle sum of 180 degrees. The relative lengths of these sides determine its specific category.

When a triangle features three sides of equal length, it is classified as an equilateral triangle. Consequently,

every interior angle in an equilateral triangle measures exactly 60 degrees.

Area of Equilateral Triangle Formula

The standard formula for calculating the area of an equilateral triangle (A) is expressed as follows:

A = (√3/4)a²

In this equation, 'a' represents the uniform length of the sides of the equilateral triangle.

To successfully find the area of an equilateral triangle, you only need to know the precise measurement of its side length.

Simply apply the formula A = (√3/4)a² to compute the area, where 'a' denotes the side length of the equilateral triangle.

What is an equilateral triangle?

In short, an equilateral triangle is defined by three sides of equal measure and three interior angles that each equal 60 degrees. 

Other key properties of an equilateral triangle include:

• The perimeter of an equilateral triangle is 3s, where 's' stands for the side length.

• The orthocenter and the centroid of the triangle occupy the exact same point.

• The median, angle bisector, and perpendicular altitude are completely identical.

Derivation of Area of the Equilateral Triangle

Now, let us walk through the step-by-step derivation of the equilateral triangle area formula.

We are already familiar with the general formula for the area of any standard triangle, which is:

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

Here, we designate height as 'h' and base as 'a'.

By adapting this foundational formula, we can easily determine the area specific to an equilateral triangle.

Consider an equilateral triangle with side length (a) and height (h).

By applying Pythagoras' Theorem to the split right-angled triangle, we establish that:

H² = P² + B² .... (ii)

Substituting the appropriate values into equation (ii) based on the geometric properties yields:

a² = h² + (a / 2)²

h² = a² - (a² / 4)

h² = (3a²  / 4)

h = ½ (√3a)

Next, substitute this derived value of "h" back into 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)

Understanding the Perimeter of an Equilateral Triangle:

Because all three sides are identical in length, the perimeter of an equilateral triangle is simply the sum of all sides, or equivalently, three times the length of one side.

The perimeter of an Equilateral Triangle = 3a,

where 'a' represents the side length.

Additionally, keep this helpful relation in mind:

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

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