Maths Equilateral Triangle

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What is an Equilateral Triangle?

If you are looking for a basic overview of equilateral triangle then it could be- A triangle in which all the sides, angles and vertex are equal can be termed as equilateral triangle. It is also known by equiangular triangle as all its angles are equal i.e. 60°. When you split the word equilateral you will encounter two meanings from it. 'Equi' which means-equivalent and 'lateral' which means sides. As already mentioned, the equilateral triangle has equal angles. Hence, all the three angles fulfills the congruence criteria, that makes it congruent to each other. And it is also a regular polygon that includes three equal sides as far as the geometry is concerned.

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Properties of an Equilateral Triangle

An equilateral triangle has some properties that prove it as a complete equiangular or equilateral triangle. Let’s explore some of the important properties of the equilateral triangle.

  •  Three sides are equal.

  •  Three angles are equal i.e 60° each.

  • A regular polygon having three equal sides.

  • Median, angle bisector and altitude are all equal in an equilateral triangle.

  • The dividing perpendicular from the angle of the vertex is parted into two equal halves i.e. 30°.

  • The area of an equilateral triangle is √3/4(a)square.

  • Perimeter of an equilateral triangle includes 3a. Where ‘A’ stands for sides.

How Is the Equilateral Triangle Different From Isosceles And Scalene Triangle?

Triangles are mainly classified into three types based on their sides. Scalene triangle, isosceles triangle and equilateral triangle. Here you are reading about equilateral triangles. But what makes equilateral triangles different from other triangles based on sides? The differentiation is:

  • Scalene Triangle: 

A triangle that’s none of the sides are equal and congruent. Also the angles vary from each other. However, their interior angles measure 180°.

  • Isosceles Triangle: 

A triangle whose any of the two sides are equal and congruent and the opposite angles of equal sides are similar to each other.

  • Equilateral Triangle: 

A triangle that’s all the sides and angles are equal and congruent to each other.


 Sides and Angles of an Equilateral Triangle

As mentioned earlier about equilateral triangles, to know deeply about equilateral triangles we must know the basic mathematical definition. An equilateral triangle is a regular polygon whose all the sides and angles are equal in length. Also they are congruent to each other. Each of the angles of the equilateral triangle measures 60°.


As far as the definition is concerned let ABC is an equilateral triangle. So side AB=BC=AC are equal and congruent. Same goes with their angles which measures 60° each and congruent too.


Equilateral Triangle Sides and Angle Congruence

We know that all the sides and angles are equal in equilateral triangle as far as the definition of math is concerned. But how will you prove that angles of the equilateral triangle are similar to each other.


Let suppose ∆ABC is an equilateral triangle. Now here you have to prove that if the sides are equal then all the interior angles are also equal.


Let, the side BC is not given to you. You have the sides AB and AC. So here you can conclude that the opposite angles of the sides AB and AC are equal. Therefore,<ABC=<ACB.


Now in another case let suppose you are not given the side AC. In this case again the opposite angles of given equal sides will be equal. Therefore, <BAC=<ACB.

Let, <ABC=<ACB be equation no.1

And <BAC=<ACB be equation no.2

Therefore,<ABC=<ACB=<BAC=X°.

 We know that the sum of the interior angles are 180°. So, x°+x°+x°=180°

                 3x=180°

                  X=180/3

                  X=60°

Therefore, you conclude that if the sides of the equilateral triangle are equal then their interior angles will also be similar. Each angle measures 60°.


Centroid of an Equilateral Triangle

An equilateral triangle consists of orthocentre, centroid, circumcentre and incentre coincide. Centroid is the centre of the circle of equilateral triangle, where height coincides with median. Ultimately centroid is the point of the triangle where all the medians interconnect.


Perimeter of Equilateral Triangle

Perimeter of an equilateral triangle is basically the sum of all given sides as all the sides of an equiangular triangle is equal as far as the definition of math is concerned.


As a whole we can conclude that the perimeter of the equilateral triangle is=sum of all sides.


Let suppose ∆ABC is an equilateral triangle whose all the sides are equal and each angle measures 60°.


Therefore, perimeter= AB+BC+CA.


Area of Equilateral Triangle

The amount of space occupied in a two dimensional plane of an equilateral triangle is called the area of equiangular or equilateral triangle. The formula goes like this:


Area of equilateral triangle:(√3/4)a2


Solved Examples


1. Find the Area of a Triangle Whose Perimeter is 18cm and Each Side Measures 6cm.

Solution:  According to the formula,area of equilateral triangle is= (√3/4)a2

          =   Side = 6cm (given)

            = √3/4(6)square

             =  √3/4×36

      = 9√3cm square.


2. Find the Perimeter of the Equilateral Triangle Whose One Side is 5cm.

Solution: We know that the perimeter of the equiangular triangle is= sum of all given sides.

Therefore, 5cm is the given and we know the equilateral triangle has all its sides equal.

So, perimeter = 5+5+5 (all sides are equal)

                  10cm


Key Points To Know About Equilateral Triangle

Before getting in touch with the main exercise, you must remember all the key points about equilateral triangle. As mentioned all the points above. If you follow all these points you will not face any problem while solving any questions about equilateral triangles. First of all,read all the properties. Get all the definitions of equiangular triangles. Find out the meaning of equilateral triangle. Memorise area and perimeter. And cover up all the key points and basics. Although I have provided you the basics.

FAQ (Frequently Asked Questions)

1. What is the Radius of the Circumscribed Circle of Equilateral Triangle?

Ans: R= a/√3

2. What is the Altitude from Any Side of the Triangle?

Ans: h= √3/2×a

3. How to Construct an Equilateral Triangle?

Ans: First of all, pull out a straight line segment and put the tip of the compass in any of the end. Now let your compass draw an arc from that point to another. Redo the same technique to the other side of the line segment. At the end join the point of intersection of the two arcs. You will be ready with an equilateral triangle.

4. If You are Given All the Angles Equal. Will it be an Equilateral Triangle?

Ans: Let ∆ABC is an equilateral triangle whose all the angles are equal i.e. <A=<B=<C that makes 180°.Your given to prove that it is an equilateral triangle.


Let, you are not given <A. You just have <B and <C. We know that if the angles of a triangle are equal then their opposite sides will also be similar. Therefore, AB=AC.


Now in the next case, let's suppose you are not familiar with <C. As you did above the same method will be applied here that is BC=AC. Therefore, AB=BC=AC. Hence, all the sides are equal and as a result given ∆ABC is an equilateral triangle.