Area of an Equilateral Triangle: Formula, Derivation & Examples

Area of an Equilateral Triangle: Understanding the area of an equilateral triangle is essential for mastering geometry problems, particularly when calculating the space enclosed by its sides on a 2D plane. In an equilateral triangle, the median, angle bisector, and altitude are identical, serving as lines of symmetry. Calculating this area is straightforward once you know the right approach. This article covers the equilateral triangle area formula, its mathematical derivation, and other key properties.

Triangles are fundamental geometric shapes categorized into three primary types based on their side lengths: scalene, equilateral, and isosceles triangles.

• A scalene triangle has no sides or angles that are equal to one another.

• An isosceles triangle features two equal sides, with the angles opposite those sides also being equal.

Area of an Equilateral Triangle

The area of an equilateral triangle represents the total surface space it occupies on a 2D plane. As the simplest form of a regular polygon, a triangle is defined as a closed geometric figure with three vertices, three sides, and three internal angles that sum to 180°. The specific classification of a triangle depends entirely on the relative lengths of these sides.

An equilateral triangle is defined by having three sides of equal length. Consequently,

each internal angle of an equilateral triangle measures exactly 60 degrees.

Area of Equilateral Triangle Formula

The mathematical formula to calculate the area (A) of an equilateral triangle is,

A = (√3/4)a²

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

To determine the area of an equilateral triangle, you only need the measurement of a single side length.

Simply plug the side length 'a' into the formula A = (√3/4)a² to find the total area.

What is an equilateral triangle?

In summary, an equilateral triangle is a polygon with three equal sides and three interior angles of 60 degrees each.

Other important properties of an equilateral triangle include:

• The perimeter is calculated as 3s, where 's' is the length of a side.

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

• The median, angle bisector, and altitude segments are all identical lines.

Derivation of Area of the Equilateral Triangle

Let’s walk through the derivation of the equilateral triangle area formula.

We begin with the standard geometric area formula for any triangle:

Area = 1⁄2 × height × base .... (i)

Where height = h and base = a.

By substituting the height relative to the side, we can define the area specifically for equilateral triangles.

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

Applying the Pythagorean Theorem to the right triangle formed by the altitude, we get:

h² + (a/2)² = a² .... (ii)

Solving for 'h' in terms of 'a':

h² = a² - (a/2)²

h² = a² - (a²/4)

h² = 3a² / 4

h = (√3/2)a

Now, substitute the value of 'h' back into equation (i):

Area = 1⁄2 × height × base

Area = 1/2 × (√3/2)a × a

Area of Equilateral Triangle = (√3/4)a²

Regarding the perimeter of an equilateral triangle:

Since all three sides are equal, the perimeter is simply the sum of the three sides, or 3 times the side length.

Perimeter = 3a,

where 'a' is the side length.

Additional useful metrics include:

• Semi-perimeter = 3a/2

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