Triangles can be grouped into three main types based on the length of their sides:

• A scalene triangle has all three sides of different lengths.
• An isosceles triangle has two sides that are the same length.
• An equilateral triangle has all three sides equal.

An equilateral triangle is a special type of triangle where not only are all the sides the same length, but all the angles are also equal — each angle measures 60 degrees. This makes it a regular polygon, meaning all its sides and angles are the same.

In this guide, we will explore the concept of equilateral triangles in detail. We’ll cover its key properties, useful formulas, and how it compares with scalene and isosceles triangles. We'll also go through a few simple solved examples to help you understand how to use these formulas in real problems.

Equilateral Triangle

An equilateral triangle has all three sides and angles equal and identical, where each of its angles is equal to 60 degrees.

Consider the figure above, triangle ABC,

Here, AB = BC = CA = “a”.

The interior angles = ∠ABC=∠BCA=∠CAB=60∘

Thus triangle ABC is an equilateral triangle.

Formula of Equilateral Triangle

In this article we are going to learn four formulas for an equilateral triangle,

• Area of an equilateral triangle.
• Height of an equilateral triangle.
• Perimeter of an equilateral triangle.
• Centroid of an equilateral triangle.

Area of an equilateral triangle

Area is the total space taken up by a flat surface or space taken up by a two dimensional object.

The area of an equilateral triangle is the area bounded by its three equal sides and is given by the formula

3√a²⁴ (unit²)

Where a = length of each side.

Perimeter of an equilateral triangle

A perimeter is defined as a closed path that outlines a two-dimensional shape or the length along a one-dimensional plane.

Perimeter is also the sum of each side of a two dimensional figure.

The perimeter of an equilateral triangle is given by the formula,

Perimeter (P) = 3a, (or a + a + a) where “a” is the length of each side.

Also, its semi perimeter is given by = 3a / 2.

Height of an equilateral triangle

We can calculate the height of an equilateral triangle by using the pythagorean theorem,

Consider triangle XYZ, where an altitude passes through vertex “X” which is the height of the given equilateral triangle.

According to the pythagoras theorem,

Hypotenuse² = base²+ height². – Equation (1)

Consider the figure above,

We have to find the height of the given triangle,

base = ½ of YZ = ½ a. (A)

Hypotenuse = a. (B),

Substituting (A) and (B) in equation (1), we get,

a² = (½a)² + h²
Where h is the height of the triangle.

So,
a² = a²/4 + h²

Now, solving for h²:
h² = a² − a²/4
h² = (3a²)/4

Taking the square root of both sides:
h = √(3a²/4)
h = (a√3)/2

Therefore, the formula for the height of an equilateral triangle is:
h = (a√3)/2

Centroid of Equilateral Triangle

The centroid of an equilateral triangle are the points in the triangle where the medians of the triangle meet.

In the above figure, for triangle ABC, point “O” is the centroid of the equilateral triangle.

It divides the medians in the ratio of 2 : 1.

Let the lengths of all sides AB = BC = AC be “a”.

The interior angles are 60 degrees.

In triangle ADB, ∠ABD = 60°
and ∠OBD = 30°

Using the sine rule in triangle ADB:
sin(60°) = AD / AB
= AD / a

√3 / 2 = AD / a
AD = (a√3) / 2

Since the median of an equilateral triangle divides the altitude in the ratio 2:1,
OD = AD / 3
OD = (a√3) / 6

Now, in triangle OBD:
∠OBD = 30°
Using the sine rule again:
sin(30°) = OD / OB
1/2 = [(a√3) / 6] / OB

Solving for OB:
OB = (a√3) / 3

Hence, for an equilateral triangle with side length “a,” the distance from the centroid to a vertex is always:
OB = (a√3) / 3

Learn about Mensuration 2D

Characteristics of an Equilateral Triangle

The following are the characteristics of an equilateral triangle

• All the sides are of equal length an identical
• All the interior angles are equal to 60 degrees.
• The centroid, the circumcenter and the orthocenter of an equilateral triangle are one and the same.
• The sum of all the interior angles in an equilateral triangle is equal to 180 degrees.

Equilateral triangle Symmetry

In this article we are going to learn about the rotation and reflection symmetry of an equilateral triangle. There are two types of symmetry that an equilateral triangle exhibits.

Rotational Symmetry

An equilateral triangle has rotational symmetry of the order 3 so that rotating it through, 120 degrees, 240 degrees and 360 degrees about its centroid, each possible arrangement we get of the vertex A, B and C and thus the triangle is analogous to the previous original form.

Reflection Symmetry

In addition to rotational symmetry an equilateral triangle also exhibits reflection symmetry. It has three lines of symmetry, passing through its three vertices and the midpoint of the side opposite to them. Across these lines the two parts of the triangle are mirror images of each other.

Properties of an Equilateral Triangle

• All three sides of an equilateral triangle are exactly the same length.
• All three angles inside the triangle are equal, and each angle measures 60 degrees.
• It is a type of regular triangle because all sides and angles are the same.
• If you draw a line from one corner (a vertex) straight down to the opposite side, it will cut that side into two equal parts. This line also splits the angle at the top into two 30-degree angles.
• In an equilateral triangle, the centroid (the point where all three medians meet) and the orthocenter (where altitudes meet) are at the exact same spot.
• The medians, angle bisectors, and altitudes from each side are all the same line in an equilateral triangle.
• The area of an equilateral triangle can be found using the formula:
  Area = (√3 × a²) ÷ 4, where a is the length of a side.
• The perimeter (total length around the triangle) is:  Perimeter = 3 × a.

Equilateral Triangle Formulas:

Difference between Scalene, Isosceles and Equilateral Triangles

Equilateral Triangle Solved Examples

Example 1. What is the area of an equilateral triangle, if the length of each of its sides is 9 cm?

Solution

Given data,
Length of each side (a) = 9 cm.

Area of an equilateral triangle,
(√3 / 4) × a²
= (√3 / 4) × 9²
= (√3 / 4) × 81
= 35.07 cm²

Area of the given equilateral triangle is 35.07 cm².

Example 2. The perimeter of an equilateral triangle is 26 cm, what is the length of each of its sides?

Solution

Given data,
Perimeter of the triangle = 26 cm.

The perimeter of an equilateral triangle is (P) = 3a,
where “a” is the length of each of its sides.

3a = 26
a = 26 / 3
a = 8.66 cm

Length of each side of an equilateral triangle is 8.66 cm.

Example 3. What would be the height and area of an equilateral triangle if the length of each side is 6 cm?

Solution

Given data,
Length of each side (a) = 6 cm

The formula for height of an equilateral triangle is:
h = (√3 / 2) × a

Height (h) = (√3 / 2) × 6
Height (h) = 5.196 cm

Area of an equilateral triangle is:
(√3 / 4) × a²

Area = (√3 / 4) × 6²
Area = (√3 / 4) × 36
Area = 15.58 cm²

Area of the given equilateral triangle is 15.58 cm².

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