Equilateral Triangle

The triangle which has all its sides equal in length is called an equilateral triangle. The word equilateral can be split into two parts: ‘equi’, which means equivalent and ‘lateral’ which means sides. It is also called an equiangular triangle because of the three angles of the triangle measure equal to 60 degrees. Hence, the three angles are congruent to each other. In geometry, it is a type of regular polygon having three equal sides. Therefore, we can also term it as a regular triangle.

Also, learn: Types Of Triangles

Table of contents:

• Definition
• Properties
• Theorem
• Area
• Perimeter
• Centroid
• Examples

Definition

As we have already discussed in the introduction, an equilateral triangle is a triangle which has all its sides equal in length. Also, the three angles of the equilateral triangle are congruent and equal to 60 degrees.

Suppose, ABC is an equilateral triangle, then, as per the definition;

AB = BC = AC, where AB, BC and AC are the sides of the equilateral triangle.

And

∠A = ∠B = ∠C = 60°

Based on sides there are other two types of triangles:

• Scalene Triangle
• Isosceles Triangle

Basic Properties

• All three sides are equal.
• All three angles are congruent and are equal to 60 degrees.
• It is a regular polygon with three sides.
• The perpendicular drawn from vertex of the equilateral triangle to the opposite side bisects it into equal halves. Also the angle of the vertex from where the perpendicular is drawn is divided into two equal angles, i.e. 30 degrees each.
• The ortho-centre and centroid are at the same point.
• In an equilateral triangle, median, angle bisector, and altitude for all sides are all the same.
• The area of an equilateral triangle is √3a²/ 4
• The perimeter of an equilateral triangle is 3a.

Theorem

If ABC is an equilateral triangle and P is a point on the arc BC of the circumcircle of the triangle ABC, then;

PA = PB + PC

Proof: For a cyclic quadrilateral ABPC, we have;

PA⋅BC=PB⋅AC+PC⋅AB

Since we know, for an equilateral triangle ABC,

AB = BC = AC

Therefore,

PA.AB = PB.AB+PC.AB

Taking AB as a common;

PA.AB=AB(PB+PC)

PA = PB + PC

Hence, proved.

Equilateral Triangle Formula

The formula for area and perimeter is given here.

Equilateral Triangle Area

The area of an equilateral triangle is the region occupied by it in a two-dimensional plane. The formula for the area of an equiangular triangle is given by:

A = √3a²/4

Let us derive the formula here:

If we see the above figure, the area of a triangle is given by;

Area = ½ x base x height

Here Base = a and height = h

Therefore,

Area = ½ x a x h ………(1)

Now, from the above figure, the altitude h bisects the base into equal halves, such as a/2 and a/2. It also forms two equivalent right-angled triangles.

So, for a right triangle, using Pythagoras theorem, we can write:

a² = h² + (a/2)²

or

h² = (a)² – (a/2)²

= 3a²/4

h = √3a/2

By putting this value in equation 1, we get;

Area = ½ x a x √3a/2

A = √3a²/4

Hence, the area of the equilateral triangle equals to √3a²/4.

Equilateral Triangle Perimeter

In geometry, the perimeter of any polygon is equal to the length of its sides. In the case of the equilateral triangle, the perimeter will be the sum of all the three sides.

Suppose, ABC is an equilateral triangle, then the perimeter of ∆ABC is;

Perimeter = AB + BC + AC

P = a + a + a

P = 3a

Where a is the length of sides of the triangle.

Centroid of Equilateral Triangle

The centroid of the equilateral triangle lies at the center of the triangle. Since all its sides are equal in length, hence it is easy to find the centroid for it.

To find the centroid, we need to draw perpendiculars from each vertex of the triangle to the opposite sides. These perpendiculars are all equal in length and intersect each other at a single point, which is known as centroid. See the figure below:

Note: The centroid of a regular triangle is at equidistant from all the sides and vertices.

Examples

Q.1: Find the area of the equilateral triangle ABC, where AB=AC=BC = 4cm.

Solution:

By the formula, we know;

Area = √3a²/4

Given a = 4cm

Hence, by putting the value we get;

Area = √3(4)²/4

A = 4√3

Q.2: Find the altitude of an equilateral triangle whose sides are equal to 10cm.

Solution:

By the formula we know,

Altitude of equilateral triangle = √3a/2

Since, a = 10cm

Hence,

h = √3 x (10/2)

h = 5√3

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