**Equilateral Triangle**

**What is an equilateral triangle**

An **equilateral triangle** is a triangle that has three sides of equal length. Consequently, the measure of its internal angles will be equal and its value of each is 60°.

The equilateral triangle is also defined as that **regular polygon of three sides and equiangular at the same time** (same angles).

According to the types of triangles, the **equilateral triangle** belongs to the class: **«according to its sides»** as well as the **isosceles triangle** and scalene triangle.

However, of all the types of triangles, the equilateral triangle is the best known and perhaps the most studied in schools because of its properties and applications.

**Properties of equilateral triangle**

We will deal with the main **properties of an equilateral triangle**, which will help us solve these types of problems.

**Property 1:**

In an equilateral triangle the notable lines: Median, Angle Bisector, Altitude and Perpendicular Bisector are equal in segment and length. See figure:

**Property 2:**

When any notable line is drawn: Angle Bisector, Altitude, Median and Perpendicular Bisector in an equilateral triangle, these divide the equilateral triangle into two congruent right triangles.

The Pythagorean theorem can be applied to any of these right triangles.

**Property 3:**

In an equilateral triangle the remarkable points: Centroid, Incentre, Circuncentre and Orthocentre coincide in the same «point» and it is fulfilled that the distance from said point to a vertex is double its distance to the base.

Let’s see:

**Property 4:**

If the Figure shown is fulfilled:

– AB = BC ; and

– m∠B= 60°

Then, when drawing AC, the ABC triangle that is formed is an **equilateral triangle**.

**Perimeter of equilateral triangle**

The perimeter of a triangle is defined as the sum of the lengths of the sides.

Then calculating the **perimeter of the equilateral triangle** will be easy, we only have to know its side and add it three times, which would be the same side multiplied by three, let’s see:

From the figure, the length of the side of the equilateral triangle is «a»:

⇒ Perimeter of equilateral triangle = a + a + a

**∴ Perimeter of Equilateral Triangle = 3a**

**Area of equilateral triangle**

The **area of an equilateral triangle (S)** is calculated from the following figure:

We know that the area of a triangle is ½(base x height). In the equilateral triangle ABC of side «a»:

**⇒ S = ½.a.h ….(1)**

Since «h» is the height of the equilateral triangle, it can be calculated in relation to the side «a» and is:

**h = a√3/2 ….(2)**

Replacing (2) in (1) we have:

**∴ Area of Equilateral Triangle = a²√3/4**

**Problems Solving**

We present a series of **equilateral triangle problems**, solved step by step, where you will be able to appreciate how these types of triangle problems are solved.

**Problem 01**

**In the figure shown the height BH measures √3m. Calculate the perimeter and area of the equilateral triangle region ABC.**

**Solving:**

From the given graph we first calculate the value of «a» (side of the triangle). We have the height of the equilateral triangle, then we apply formula:

h = a√3/2 ; where: h = √3m

⇒ a = 2m

**i) Calculation of the Perimeter:** according to the theory the perimeter is equal: 3.a

⇒ Perimeter ΔABC = 3a = 3(2m)

**∴ Perimeter ΔABC = 6m**

**(ii) Calculation of the area:** applying the formula of the area of equilateral triangle:

A = a²√3/4

⇒ A = 2²√3/4

**∴ Area ΔABC = √3m²**

Excellent subject. I like geometry

A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.

I loved 😍