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How to find the perimeter of an equilateral triangle?

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  • Last Updated : 10 Nov, 2021

An Equilateral triangle is a triangle in which all three sides are equal and angles are also equal. The value of each angle of an equilateral triangle is 60 degrees therefore, it is also known as an equiangular triangle. The equilateral triangle is considered as a regular polygon or a regular triangle as angles are equal and sides are also equal.

For instance, in the triangle, ABC are equal i.e.

AB = BC = CA = a units.

Also, ∠A, ∠B and, ∠C = 60°

Properties of Equilateral Triangle

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

Perimeter of Equilateral Triangle

Semi Perimeter of an Equilateral Triangle

Semi Perimeter of an Equilateral Triangle = \frac{Perimeter\ of\ an\ Equilateral\ Triangle}{2}

Let us assume a to be the side of an equilateral triangle. 

In other words, we have, 

Semi Perimeter of an Equilateral Triangle = \frac{3a}{2}

Perimeter of an Equilateral Triangle when the area is given 

Let us assume a to be the side of an equilateral triangle. 

Perimeter of an equilateral triangle can be computed using its area, which is given by, 

Area=\frac{\sqrt3}{4}a^2

Now,

We know, 

Perimeter of an equilateral triangle = Side + Side + Side

Perimeter of an equilateral triangle, P is given by = 3 × a

Therefore, the values a can be replaced by P/3.

Area=\frac{\sqrt3}{4}(\frac{P}{3})^2

Perimeter of an Equilateral Triangle when Altitude is given

The perimeter of an equilateral can be calculated when the altitude (height) of the triangle is given. 

We have, 

Height of an Equilateral Triangle = \frac{\sqrt3}{2}a

Upon substituting the values of the perimeter of the equilateral triangle, we have, 

Perimeter of an equilateral triangle = Side + Side + Side

Perimeter of an equilateral triangle = 3 × a

Therefore, 

Height (or Altitude ) = \frac{\sqrt3}{2}(\frac{P}{3})

Sample Questions

Question 1. Calculate the perimeter of an equilateral triangle if the side of the triangle is 30√3 cm.

Solution:

Here we have to find the perimeter of an equilateral triangle 

We are given that the side of the equilateral triangle is 30√3 cm

As we know that

Formula for perimeter of an equilateral triangle

Perimeter of an equilateral triangle = Side + Side + Side

Perimeter of an equilateral triangle = 3 × a

where a is side of an equilateral triangle

Perimeter of an equilateral triangle = 3 × 30√3

Perimeter of an equilateral triangle = 90√3 cm

Therefore,

Perimeter of an equilateral triangle is 90√3 cm.

Question 2. If the side of an equilateral triangle is 90 m, then find the perimeter and semi-perimeter of the triangle?

Solution:

Here we have to find the perimeter and semi-perimeter of an equilateral triangle,

First finding the perimeter of an equilateral triangle

We are given that the side of the equilateral triangle is 90 m

Formula for the perimeter of an equilateral triangle

Perimeter of an equilateral triangle = Side + Side + Side

The perimeter of an equilateral triangle = 3 × a

Substituting the value of a in the formula

Perimeter of an equilateral triangle = 3 × 90

Perimeter of an equilateral triangle = 270 m

Further finding the semi-perimeter

Formula for semi-perimeter of an equilateral triangle = \frac{3a}{2}

Where a is the side of an equilateral triangle

Substituting value of a in the formula

Semi-perimeter of an equilateral triangle = \frac{3\times90}{2}

Semi-perimeter of an equilateral triangle = 135 m

Therefore,

Perimeter of an equilateral triangle is 270 m and semi-perimeter of an equilateral triangle is 135 m.

Question 3. Consider that the area of an equilateral triangle is 100√3 cm2, Then calculate its perimeter?

Solution:

Here we have to find the perimeter of equilateral triangle using its area

Formula for equilateral triangle area = \frac{\sqrt3}{4}a^2

Area of equilateral triangle = 100√3

100√3= \frac{\sqrt3}{4}a^2

a2100\sqrt3\times\frac{4}{\sqrt3}

a = √400

a = 20

Therefore,

Side of the equilateral triangle is 20 cm

Now further finding perimeter of equilateral triangle

Perimeter of equilateral triangle = side + side + side = 3a

Perimeter of equilateral triangle = 3 × 20

Perimeter of equilateral triangle = 60 cm

Question 4. Find the perimeter of an Equilateral triangle if the height of the triangle is 35√3 m.

Solution:

Here we have to find the Perimeter of the equilateral triangle with the height 35√3 m

Formula for calculating perimeter using height is given below

Height = \frac{\sqrt3}{2}a

Here a is the side of the equilateral triangle

35√3 = \frac{\sqrt3}{2}a

a = 35\sqrt3\times\frac{2}{\sqrt3}

a = 70 m

Now further finding perimeter of the equilateral triangle

Perimeter of equilateral triangle = side + side + side = 3a

Perimeter of equilateral triangle = 3 × 70

Perimeter of equilateral triangle = 210 m

Question 5. If the side of an equilateral triangle is 23 cm, then find the perimeter and height of the equilateral triangle?

Solution:

Here we have to find the perimeter of an equilateral triangle

We are given that the side of the equilateral triangle is 23 cm

As we know that

Formula for the perimeter of an equilateral triangle

Perimeter of an equilateral triangle = Side + Side + Side

Perimeter of an equilateral triangle = 3 × a

where a is side of an equilateral triangle

Perimeter of an equilateral triangle = 3 × 23

Perimeter of an equilateral triangle = 69 cm

Further finding the height of the triangle

Height = \frac{\sqrt3}{2}a

Here a is the side of the equilateral triangle

Substituting the value of a in the formula

Height = \frac{\sqrt3}{2}\times23

Height = 11.5√3

Therefore,

Perimeter of an equilateral triangle is 69 cm and the height of an equilateral triangle is 11.5√3 cm.


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