Perimeter of Equilateral Triangle

The perimeter of an equilateral triangle is the total length of its boundary. In geometry, an equilateral triangle is a triangle in which all three sides are equal. The total length of the boundary of a triangle can be calculated by adding the length of all its sides. In this article, we will learn how to calculate the perimeter of an equilateral triangle and its formula with the help of solved examples.

The perimeter of an equilateral triangle is the sum of its three sides. A triangle is considered to be equilateral if it has the following basic properties:

• All three sides are equal.
• All three angles are equal to 60°

In the figure given below, the sides of the triangle PQR are equal, i.e., PQ = QR = RP. Along with this, the angles of the triangle are also equal. Therefore, this is an equilateral triangle. Now, the perimeter of an equilateral triangle = 3a (where 'a' is the side of an equilateral triangle)

The basic formula that is used to calculate the perimeter of an equilateral triangle is: P = 3a, where 'a' represents one side of the triangle. Since all the three sides of an equilateral triangle are equal, the sum becomes a + a + a = 3a.

A few other formulas related to the equilateral triangle are as follows:

• Sometimes, when the sides of an equilateral triangle are given, we need to find its height, then we use the formula: Height of an Equilateral Triangle = (√3a)/2
• In a few cases, we need to find the semi-perimeter of an equilateral triangle. Semi perimeter is half the perimeter and is calculated by the formula, Semi-perimeter = (a + a + a)/2 = 3a/2

We know that the formula for the perimeter of an equilateral triangle is 3a, where a = side of an equilateral triangle. Now, let us learn how to apply the formula for the perimeter of an equilateral triangle.

Example: Find the perimeter of an equilateral triangle with side 9 units. What is the semi-perimeter of this triangle?

Solution:

Given: Side of equilateral triangle = a = 9 units. The perimeter of an equilateral triangle = 3a. Substituting the value of a = 9 in the formula = 3 × 9 = 27 units

Semi-Perimeter of an equilateral triangle = 3a/2. Substituting the value of a = 9 in the formula = (3 × 9)/2 = 27/2 = 13.5 units

Note: Semi-Perimeter of an equilateral triangle is half of the perimeter of an equilateral triangle.

Perimeter of Equilateral Triangle when Area is Given

The perimeter of an equilateral triangle can be calculated if we know the area of the triangle. In this case, we first need to find the length of the sides with the help of which the perimeter can be calculated.

Example: If the area of an equilateral triangle is 56 square units, let us find its perimeter.

Solution: We know that Area of an equilateral triangle = (a²√3)/4. So, the side can be calculated by substituting the value of area in the formula.

Area = (a²√3)/4

56 = (a²√3)/4

a² = (56 × 4)/√3

a = 11.37 units

Now, the perimeter of the equilateral triangle can be calculated, P = 3a = 3 × 11.37 = 34.11 units.

Perimeter of Equilateral Triangle when Altitude is Given

The perimeter of an equilateral can be calculated when the altitude (height) of the triangle is given. In this case, we can find the side length of the triangle with the help of the formula: Height of an Equilateral Triangle = (√3a)/2. After finding the side length, the perimeter of the equilateral triangle can be easily calculated.

Example: If the height of an equilateral triangle is given as 6 units, find its perimeter.

Solution: We know that the formula for the height of an equilateral triangle = (√3a)/ 2. So, the side length can be calculated by substituting the value of height in the formula.

Height = (√3a)/2

6 = (√3a)/2

a = (6 × 2)/√3

a = 6.92 units

Now, the perimeter of the equilateral triangle can be calculated, P = 3a = 3 × 6.92 = 20.76 units.

☛ Related Articles

• Perimeter of Right-angled Triangle
• Perimeter of Isosceles Triangle
• Perimeter of Scalene Triangle
• Perimeter of a Triangle
• Area of Equilateral Triangle
• Equilateral Triangle Formulas

What is the Perimeter of an Equilateral Triangle?

The total length of the boundary of an equilateral triangle is called its perimeter. The perimeter of an equilateral triangle can be calculated if the length of its side is known. For example, if one side of an equilateral triangle is 5 units, the perimeter = 3 × side = 3 × 5 = 15 units.

How to Find the Perimeter of Equilateral Triangle?

To find the perimeter of an equilateral triangle, its side length should be known. Since all the sides of an equilateral triangle are equal in length, the perimeter can be easily calculated with the length of one of its sides. This value can be substituted in the formula for the perimeter of an equilateral triangle, P = 3 × a; where 'a' is the side length of the triangle.

How to Find the Perimeter of Equilateral Triangle when Area is Given?

If we know the area of an equilateral triangle, we can easily find its perimeter. In this case, we first need to find the length of the sides with the help of which the perimeter can be calculated. For example, if the area of an equilateral triangle is 48 square units, let us find its perimeter. We know that Area of an equilateral triangle = (a²√3)/4. Now, the side length can be calculated by substituting the value of area in the formula. 48 = (a²√3)/4. After solving this, we get 'a' = 10.53. Now, using this side length, the perimeter can be calculated. Perimeter = 3a = 3 × 10.53 = 31.59 units.

How to Find the Perimeter of Equilateral Triangle when Altitude is Given?

The perimeter of an equilateral triangle can be calculated when the altitude (height) of the triangle is given. Here, we can find the side length of the triangle with the help of the formula: Height of an Equilateral Triangle = (√3a)/ 2. After finding the side length, the perimeter of the equilateral triangle can be calculated. For example, if the height of an equilateral is given as 8 units, let us find the perimeter. We know that the height of an Equilateral Triangle = (√3a)/ 2. Now, let us calculate the side length by substituting the value of height in the formula. 8 = 3a/2. So, a = 5.3 units. Now, using this side length, the perimeter can be calculated. Perimeter = 3a = 3 × 5.3 = 15.9 units.

Find the Perimeter of the Equilateral Triangle with side 9 cm.

If the side of an equilateral triangle is 9 cm, the perimeter can be calculated with the help of the formula, perimeter = 3a, where a = side length. Let us substitute the value of 'a' in the formula. P = 3a = 3 × 9 = 27 cm. Therefore, the perimeter of the equilateral triangle with side 9 cm is 27 cm.

Is an Equilateral Triangle a Regular Polygon?

Yes, an equilateral triangle is considered to be a regular polygon because:

• It is a triangle in which all the angles are equal.
• All the sides are equal in length.

How many Sides does an Equilateral Triangle have?

An equilateral triangle has three sides. The length of each side of an equilateral triangle is equal.

What is the Sum of all the Angles of an Equilateral Triangle?

The sum of all the angles of an equilateral triangle is equal to 180°. Each angle measures 60°. Accordingly, 60°+ 60°+ 60° = 180 degrees.

How many Angles does an Equilateral Triangle have?

An equilateral triangle has three angles. All the angles are equal to 60°. The sum of these angles is equal to 180°.

What is the Formula of Equilateral Triangle?

The basic formula that is used to calculate the perimeter of an equilateral triangle is, P = 3a, where 'a' represents one side of the triangle. The formula that is used to find the area of an equilateral triangle is, Area of equilateral triangle = (a²√3)/4, where 'a' is one side of the triangle.