Equilateral Triangle: Properties, Formula & Examples

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An equilateral triangle is a triangle with all three sides of equal length. A triangle is a plane figure formed by the three non-parallel line segments. It is named on the basis of the length of the sides and the measure of their angles.

An equilateral triangle is a special case of an isosceles triangle.
In these triangles, all angles are congruent to each other.
The value of each angle is equal to 60 degrees.
Equilateral triangles are also referred to as regular triangles.
It is a type of regular polygon.
Traffic Board signs are real life examples of an equilateral triangle.
The word equilateral is derived from a combination of two words: equity, which refers to equivalent, and lateral, which refers to sides.

Read More: Isosceles Triangle Theorems

Table of Content

What is an Equilateral Triangle?
Properties of an Equilateral Triangle
Area of an Equilateral Triangle
Perimeter of an Equilateral Triangle
Height of an Equilateral Triangle
Equilateral Triangle Formula
Things to Remember
Sample Questions

Key Terms: Equilateral Triangle, Triangles, Area, Perimeter, Angles, Sides, Line Segments, Polygon, Isosceles Triangle, Height of an Equilateral Triangle

What is an Equilateral Triangle?

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An Equilateral Triangle is a type of triangle whose all sides are equal. There are three edges and three vertices in the triangle. Since the sides of an equilateral triangle are equal, the three angles of the triangle are also equal. All three angles are also congruent to each other, and each angle measures 60 degrees.

Every centre of the equilateral triangle coincides with the centroid.
In these triangles, the circumcentre, incentre and orthocentre lie in the same plane.
The incentre of the triangle coincides with nine point centre.
The racks used in a pool table are an example of an equilateral triangle.

Solved Example of an Equilateral Triangle

Example: Calculate the area of an equilateral triangle whose sides are 30 inches.
Solution: Area of equilateral triangle = √3a² / 4, where a is the side
Given, a = 30 inches
Therefore, area = √3×30×30 / 4
Area of an equilateral triangle = 225√3 square inches

Read More:

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Quadrilateral Formula	Trapezoid Formula	Isosceles triangle
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Properties of an Equilateral Triangle

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A triangle is named on the basis of the length of the sides and the measure of their angles. Some of the properties of an equilateral triangle are as follows:

An equilateral triangle has all sides equal.
All angles of the triangle are equal.
It is also called equiangular.
The equilateral triangle is a regular polygon with three equal sides.
The region occupied by the two-dimensional space inside the triangle is called the area.
The area of the triangle is equal to (√3/4) (side)².
The boundary of a triangle is called the perimeter.
The perimeter of an equilateral triangle is equal to side 3a.
The centroid and ortho-center lie on the same point.
The median, angle bisector, and perpendicular are same.

Read More: Area of Rectangle

Area of an Equilateral Triangle

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The region occupied by the two-dimensional space inside the triangle is called the area of an equilateral triangle.

Area = (√3/4) (side)²

In an equilateral triangle, all the sides of a triangle are equal so to find the area of an equilateral triangle all you have to do is multiply the square of the sides with √3/4.

Solved Example of Area of an Equilateral Triangle

Example: Find the area of a triangle with all sides equal and where each side measures 4 cm.

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The area of an equilateral triangle is equal to (√3/4) (side)² .

Area= √3/4 *4*4

= 4√3 cm².

Read More: Construction Formula

Derivation of Area of an Equilateral Triangle

Let an equilateral triangle have a side = a and height =h .

Now, Area of Triangle = ½ × base × height
Here, base = a, and height = h
Using Pythagoras Theorem.
a² = h² + (a/2)²
⇒ h² = a² – (a²/4)
⇒ h² = (3a²)/4
Or, h = ½(√3a)
Now, put the value of “h” in the area of the triangle equation.
Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a)

Perimeter of an Equilateral Triangle

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The boundary of a triangle is called the perimeter. Perimeters are used to find the shape of a fence for various purposes in our daily life. It is the sum of all sides of a triangle. In an equilateral triangle, all the sides of a triangle are equal so to find the perimeter of an equilateral triangle all you have to do is multiply the side by 3.

Solved Example of Perimeter of an Equilateral Triangle

Example: Find the perimeter of a triangle with all sides equal and where each side measures 4 cm.

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The perimeter of an equilateral triangle is equal to side *3.

Perimeter = side *3 = 4*3= 12 cm.

Read More: Trapezoids

Height of an Equilateral Triangle

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Height of an equilateral triangle is an altitude drop from the apex of the triangle to the base. It is also known as altitude of an equilateral triangle. It is calculated by pythagoras theorem. The height of an equilateral triangle is expressed as:

h = √3a/2

Solved Example of Perimeter of an Equilateral Triangle

Example: Find the altitude of an equilateral triangle whose sides are equal to 12 cm.

Solution: By the formula, we know,

Height of an equilateral triangle = √3a/2

Since, a = 12 cm

Hence,

h = √3 x (12/2)

h = 6√3

Read More:

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Area of Parallelogram	Formula Of Perimeter Shapes	Area of a Triangle
Surface Area of a Cylinder Formula	Sphere Formula	Height and Difference

Equilateral Triangle Formula

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The important formula for an equilateral triangle are as follows:

Category	Formula
Area of an Equilateral Triangle	A = (√3/4) (side)²
Perimeter of an Equilateral Triangle	P = 3a
Height of an Equilateral Triangle	h = √3a/2

Read More: Types of polygons

Things to Remember

An equilateral triangle consists of all sides equal to one another.
All sides of triangle are equal.
All angles of are congruent to each other and are equal to 60 degrees.
The area of an equilateral triangle is equal to (√3/4) (side)²
The perimeter of the triangle is given by the formula 3a.

Read More: Diagonal

Sample Questions

Ques: What is the area of an equilateral triangle whose perimeter is 15 cm. (2 marks)

Ans. The perimeter of an equilateral triangle is equal to side *3.

Here the perimeter is given as 15 cm.

So the side measures 5 cm each.

Now the area of an equilateral triangle is equal to (√3/4) (side)².

So area of an equilateral triangle = 25(√3/4) cm².

Ques: Find the area of a triangle with all sides equal and where each side measures 16 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The area of an equilateral triangle is equal to (√3/4) (side)² .

Area= √3/4 *16*16

= 64√3 cm².

Ques: Find the perimeter of a triangle with all sides equal and where each side measures 14 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The perimeter of an equilateral triangle is equal to side *3.

Perimeter = side *3 = 14*3= 42 cm

Ques: What is the area of an equilateral triangle whose perimeter is 18 cm. (3 marks)

Ans. The perimeter of an equilateral triangle is equal to side *3.

Here the perimeter is given as 18 cm.

So the side measures 6 cm each.

Now the area of an equilateral triangle is equal to (√3/4) (side)².

So the area of an equilateral triangle = 36(√3/4) cm².

= 9√3 cm².

Ques: What are the properties of an equilateral triangle. (2 marks)

Ans. The main properties of the equilateral triangle can be described as follows:

It is a regular polygon with three sides. All three sides are equal in length.
All three angles are congruent and equal to 60º.

Ques: What is the sum of angles in an equilateral triangle and What is the height of the Equilateral Triangle. (2 marks)

Ans. The sum of all the angles of the equilateral triangle is 180 degrees. Formula to calculate the height of an equilateral triangle is given as: Height of an equilateral triangle, h = (√3/2)a, where a is the side of the equilateral triangle.

Ques: What is the area of an equilateral triangle whose perimeter is 18 cm. (2 marks)

Ans. The perimeter of an equilateral triangle is equal to side 3a.

Here the perimeter is given as 18 cm.

So the side measures 6 cm each.

Now the area of an equilateral triangle is equal to (√3/4) (side)².

So area of an equilateral triangle = 36(√3/4) cm².

Ques: Find the area of a triangle with all sides equal and where each side measures 20 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The area of an equilateral triangle is equal to (√3/4) (side)² .

Area= √3/4 *20*20

= 25√3 cm².

Ques: Find the perimeter of a triangle with all sides equal and where each side measures 27 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The perimeter of an equilateral triangle is equal to side *3.

Perimeter = side *3 = 27*3= 81 cm

Ques: What is the area of an equilateral triangle whose perimeter is 45 cm. (3 marks)

Ans. The perimeter of an equilateral triangle is equal to side 3a.

Here the perimeter is given as 45 cm.

So the side measures 15 cm each.

Now the area of an equilateral triangle is equal to (√3/4) (side)².

So the area of an equilateral triangle = 225(√3/4) cm².

= 22√3 cm².

Ques: What are the perimeter and semi perimeter of an equilateral triangle with side measurement as 15 units. (2 marks)

Ans. Perimeter of equilateral triangle = 3a and semi-perimeter of an equilateral triangle = 3a/ 2, where a is the side.

Given a = 15 units.

Therefore, the perimeter of equilateral triangle = 3 × 15 = 45 units and Semi-perimeter of an equilateral triangle = 45/2 = 22.5 units.

Ques: Find the area of the equilateral triangle ABC, where AB=AC=BC = 8 cm. (2 marks)

Ans. By the formula, we know;

Area = √3a²/4

Given a = 8 cm

Hence, by putting the value we get;

Area = √3(8)²/4

A = 4√3

Ques: Find the area of a triangle with all sides equal and where each side measures 80 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The area of an equilateral triangle is equal to (√3/4) (side)² .

Area= √3/4 *80*80

= 400√3 cm².

Ques: Find the perimeter of a triangle with all sides equal and where each side measures 71 cm. (2 marks)

Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The perimeter of an equilateral triangle is equal to side *3.

Perimeter = side *3 = 71*3= 81 cm

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Equilateral Triangle: Properties, Formula & Examples

What is an Equilateral Triangle?

Solved Example of an Equilateral Triangle

Properties of an Equilateral Triangle

Area of an Equilateral Triangle

Solved Example of Area of an Equilateral Triangle

Derivation of Area of an Equilateral Triangle

Perimeter of an Equilateral Triangle

Solved Example of Perimeter of an Equilateral Triangle

Height of an Equilateral Triangle

Solved Example of Perimeter of an Equilateral Triangle

Equilateral Triangle Formula

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

4.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

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Equilateral Triangle: Properties, Formula & Examples

What is an Equilateral Triangle?

Solved Example of an Equilateral Triangle

Properties of an Equilateral Triangle

Area of an Equilateral Triangle

Solved Example of Area of an Equilateral Triangle

Derivation of Area of an Equilateral Triangle

Perimeter of an Equilateral Triangle

Solved Example of Perimeter of an Equilateral Triangle

Height of an Equilateral Triangle

Solved Example of Perimeter of an Equilateral Triangle

Equilateral Triangle Formula

Things to Remember

Sample Questions

CBSE X Related Questions

1.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

4.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

4.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

5.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

6.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.