Open In App

Equilateral Triangle

Improve
Improve
Like Article
Like
Save
Share
Report

Equilateral Triangle is a triangle with all three sides and all three angles equal. As we know a triangle has three angles and three sides, thus in an equilateral triangle all three sides and all three angles are equal. Hence, it is also called a regular triangle.

The word equilateral is made of two words “equi” and “lateral” where equi means equal and lateral means side. Thus equilateral triangle means a triangle with equal sides. In this article, we learn about equilateral triangles, their properties, their formulas, and others in detail.

What is an Equilateral Triangle?

An equilateral triangle is defined as a triangle that has three equal sides and three equal angles. Since the three sides of an equilateral triangle are equal, it is considered a regular polygon.

In the figure given below, ∆ABC is an equilateral triangle with equal sides that measures “a” unit, i.e., AB = BC = AC = a. 

Equilateral Triangle

Equilateral Triangle Angles

We can also observe that the three interior angles of the ∆ABC measure 60°, i.e., ∠A = ∠B = ∠C = 60°.  This can be proved by using the angle sum property of a triangle.

We know that sum of all angles of a triangle is 180°

Now, ∠A + ∠B + ∠C = 180°…(i)

 In an equilateral triangle,

∠A = ∠B = ∠C

⇒ ∠A + ∠A + ∠A = 180° [Form (i)]

⇒ 3∠A = 180°

⇒ ∠A = 60°

Then, ∠A = ∠B = ∠C = 60°

Equilateral Triangle Formulas

Equilateral Triangle Formula

Each Interior Angle of Equilateral Triangle

60°

Each Exterior Angle of Equilateral Triangle

120°

Perimeter of Equilateral Triangle

3 × Sides

Height of Equilateral Triangle

√3/2 × (Side)

Area of Equilateral Triangle

√3/4 × (Side)2

Equilateral Triangle Shape

Shape of the equilateral triangle resembles a regular polygon with three sides. It is the simplest regular shape and is widely observed in our surroundings. Various equilateral shapes objects are,

  • Tiles
  • Clocks
  • Nachos
  • Sweets, etc

The image added below shows some equilateral triangle objects

Examples of Equilateral Triangle

An equilateral triangle has all three sides and all three and equal. In the equilateral triangle ABC,

  • AB = BC = CA
  • ∠A = ∠B = ∠C = 60°

Properties of Equilateral Triangles

Some important characteristics of an equilateral triangle are,

  • All three side lengths of an equilateral triangle always measure the same.
  • The three interior angles of an equilateral triangle are congruent and equal to 60°.
  • According to the angle sum property, the sum of the interior angles of an equilateral triangle is always equal to 180°.
  • Equilateral triangles are considered regular polygons since their three side lengths are equal.
  • The perpendicular drawn from any vertex of an equilateral triangle bisects the opposite side into two halves. The perpendicular also bisects the angle at the vertex from which it is drawn into 30° each.
  • In an equilateral triangle, the orthocenter and centroid are at the same point.
  • Median, Angle Bisector and Altitude for all sides of an equilateral triangle are the same.
  • Area of an Equilateral Triangle is √3/4 a2, where “a” is the side length of the triangle.
  • Perimeter of an Equilateral Triangle is 3a, where “a” is the side length of the triangle.

Equilateral Triangle Theorem

Equilateral triangle theorem states that, 

“For any equilateral triangle ABC, if P is any point on the arc BC of the circumcircle of the triangle ABC, then PA = PB + PC”

Proof: 

In cyclic quadrilateral ABPC, we have,

PA⋅BC = PB⋅AC + PC⋅AB

As ABC is an equilateral triangle,

AB = BC = AC

Thus,

PA.AB = PB.AB + PC.AB

Simplifying,

PA.AB = AB(PB + PC)

PA = PB + PC

Hence, proved.

Equilateral Triangle Formulas

Now to measure various aspects of the equilateral triangle we use three common equilateral triangle formulas that are,

  • Equilateral Triangle Height
  • Equilateral Triangle Perimeter
  • Equilateral Triangle Area
  • Equilateral Triangle Centroid

Let’s learn about them in detail below in this article.

Height of Equilateral Triangle

The height or altitude of an equilateral triangle is equal to √3a/2, where “a” is the side length of the triangle. It can be determined by using the Pythagorean formula. In the figure given below, ∆ABC is an equilateral triangle with equal sides that measures “a” unit. We can observe that a perpendicular is drawn from the vertex A to the opposite side BC, bisecting it into two halves at point D. ABD and ADC are two equal right angles. The length of AD is the height of the given triangle ∆ABC.

Equilateral Triangle Height

Height of an Equilateral Triangle = √3a/2

where “a” is the side length of the triangle.

Perimeter of Equilateral Triangle

The perimeter of an equilateral triangle is equal to the sum of its three side lengths. We know that all three sides of an equilateral triangle are equal. So, the perimeter of an equilateral triangle is 3a, where “a” is the side length of the triangle.

 In the figure given above, ∆ABC is an equilateral triangle with equal sides that measures “a” unit. 

So, the perimeter of an equilateral triangle (P) = (AB + BC + AC) units

(P) = a + a + a = 3a units

Perimeter of an equilateral triangle (P) = 3a units

where “a” is the side length of the triangle.

Learn more about, Perimeter of a Triangle

Area of Equilateral Triangle

The total region bounded by the three sides of a triangle in a two-dimensional plane is known as the area of a triangle. The area of an equilateral triangle is √3/4 a2, where “a” is the side length of the triangle.

Area of an Equilateral Triangle = √3/4 a2

where “a” is the side length of the triangle.

Learn more about, Area of an Equilateral Triangle

Area of Equilateral Triangle using Heron’s Formula

We know that the area of a triangle can be calculated using Heron’s formula if all its three side lengths are given.  In the figure given above, ∆ABC is an equilateral triangle with equal sides that measures “a” unit. 

So, AB = BC = CA = a

We know that,

Area of Triangle = √{s(s-a)(s-b)(s-c)}

where,

  • s is Semi-Perimeter
  • 9s = (a + b + c)/2
  • a, b, and c are Side Lengths of Triangle

Here, a = b = c = a

So, s = (a + a + a)/2 = 3a/2

Now, substitute the values in the formula.

A = √{3a/2(3a/2-a)(3a/2-a)(3a/2-a)}

A = √{(3a4)/(4)2}

A = (√3/4) a2

Hence,

Area of Equilateral Triangle = √3/4 a2

Learn more about, Heron’s Formula

Centroid of Equilateral Triangle

Centroid of the triangle also called the centre of the triangle is a point which is at the centre of the triangle. This point is equidistant from all three vertices of the triangle. For an equilateral triangle, as all the sides are equal in length, it is easy to find the centroid for it.

If we draw perpendicular from all the vertices of the equilateral triangle to their opposite sides the point where they all meet is the centroid of the equilateral triangle.

We know that the meeting point of all three perpendiculars of the triangle is called the orthocentre of the triangle. Thus, for an equilateral triangle, the Centroid and Orthocentre are the same points.

For any equilateral triangle ABC its centroid is denoted using point A in the image added below,

Centroid-of-Triangle

In equilateral triangle with length “a” the distance from the centroid to the vertex is equal to √(3a/3)

Circumcenter of Equilateral Triangle

The centre of the circle passing through all three vertices of the triangle is called the circumcentre of the triangle. It is calculated by taking the intersection of any two perpendicular bisectors of the triangle.

Note: In an Equilateral triangle, the incenter, orthocenter and centroid all coincide with the circumcenter of the equilateral triangle.

Equilateral Triangle Symmetry

An equilateral triangle has two types of symmetry that includes,

  • Rotational Symmetry
  • Reflection Symmetry

Now let’s learn about both of them in detail.

Rotational Symmetry

In an equilateral triangle we have rotational symmetry of order 3 and rotationg it 120 degrees, 240 degrees, and 360 degrees results in the symmetrical triangle.

Reflection Symmetry

In an equilateral triangle we also have reflection symmetry. It has three lines of reflection symmetry and all its median acts as line of reflection symmetry.

Difference Between Scalene, Isosceles, and Equilateral Triangles

Major differences between Scalene Triangle, Isosceles Triangle and Equilateral Triangle is added in the table below,

Scalene vs Isosceles vs Equilateral Triangles

 Scalene Triangle 

 Isosceles Triangle 

 Equilateral Triangle 

All three side lengths of a scalene triangle are always unequal. There will be at least two equal side lengths in an isosceles triangle. All three side lengths of an equilateral triangle always measure the same.
All three interior angles of a scalene triangle are always unequal. The interior angles opposite the equal sides of an isosceles triangle are equal. The three interior angles of an equilateral triangle are congruent and equal to 60°.
Scalene Triangle  Isosceles Triangle Equilateral Triangle

People Also Read:

Equilateral Triangle Examples

Example 1: Determine the area of an equilateral triangle whose side length is 10 units.

Solution:

Given,

  • Side length (a) = 10 units

We know that,

Area of Equilateral Triangle = √3/4 a2

A = √3/4 × (10)2

⇒ A = √3/4 × 100 

⇒ A = 25√3 square units ≈ 43.301 square units

Hence, the area of the given equilateral triangle is approximately equal to 43.301 square units.

Example 2: Determine the height of an equilateral triangle whose side length is 8 cm.

Solution:

Given,

  • Side length (a) = 8 cm

We know that,

Height of Equilateral Triangle = √3a/2

⇒ H = √3/2 × 8

⇒ H = 4√3 cm

⇒ H ≈ 6.928 cm

Hence, the height of given equilateral triangle is approximately equal to 6.928 cm.

Example 3: Determine the perimeter of an equilateral triangle whose side length is 13 cm.

Solution:

Given,

  • Side length (a) = 13 cm

We know that,

Perimeter of Equilateral Triangle (P) = 3a units

⇒ P = 3 × 13 = 39 cm.

Hence, the perimeter of the given equilateral triangle is 39 cm.

Example 4: What is the area of an equilateral triangle if its perimeter is 36 cm?

Solution:

Given,

Perimeter of Equilateral Triangle (P) = 36 cm

We know that,

Perimeter of Equilateral triangle (P) = 3a units

⇒ 3a = 36

⇒ a = 36/3 = 12 cm

We know that,

Area of Equilateral Triangle = √3/4 a2

⇒ A = √3/4 × (12)2

⇒ A = √3/4 × 144

⇒ A = 36√3 sq. cm

Hence, Area of the given equilateral triangle is 36√3 sq. cm.

Practice Questions on Equilateral Triangle

Some practice question on Equilateral Triangle are,

Q1: Find the area of an equilateral triangle if its perimeter is 48 cm.

Q2: If the side of triangle is 19 cm then find the area of that equilateral triangle.

Q3: Find the area of an equilateral triangle if its perimeter is equal to the perimeter of square of side 6 cm.

Q4: Find the perimeter of equilateral if its side is equal to side of square with area 64 m2.

Summary: Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides and all three angles are equal, making it a regular polygon. This means each side is the same length, which we often represent as “a,” and each of the interior angles is 60 degrees. Because of its symmetry, equilateral triangles have several unique properties. For example, the height, which can be found by drawing a perpendicular line from one vertex to the opposite side, is calculated using the formula √3/2 times the side length. This symmetry also means that the triangle’s area can be found with the formula √3/4 times the square of the side length, and its perimeter is simply three times the side length. Equilateral triangles are seen in various objects around us, like tiles, clocks, and sweets, due to their balanced and harmonious shape. They’re important in geometry for their properties and theorems, such as the equilateral triangle theorem, which relates to the lengths of lines drawn within the triangle’s circumcircle. Overall, equilateral triangles are fascinating for their equal sides and angles, making them a key topic in the study of geometry.

FAQs on Equilateral Triangle

What is Equilateral Triangle?

A triangle with all three sides and all three angles equal is called an equilateral triangle. 

What is Equilateral Triangle Altitude Formula?

Altitude(A) of an equilateral triangle is measured using the formula, A = √3a/2

What is Perimeter of Equilateral Triangle Formula?

Perimeter of an equilateral triangle is equal to the sum of its three side lengths. We know that all three sides of an equilateral triangle are equal. So, the perimeter of an equilateral triangle is 3a, where “a” is the side length of the triangle.

What is Area of Equilateral Triangle Formula?

Area of an equilateral triangle is √3/4 a2, where “a” is the side length of the triangle.

Is Equilateral Triangle a Regular Polygon?

Yes, equilateral triangle is a regular polygon. In fact, is the simplest regular polygon. A regular polygon is a polygon in which all the sides are equal, and in equilateral triangle all the sides are equal. Hence it is a regular polygon.

How many Sides and Angles does an Equilateral Triangle have?

An equilateral triangle has three equal sides and three equal angles. The measure of each angle in an equilateral triangle is 60°



Last Updated : 26 Mar, 2024
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads