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How to find the length of one side of an equilateral triangle?

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  • Last Updated : 20 Nov, 2021

In mathematics or in daily lives geometry and geometrical shapes always play an essential role. Starting from a simple tray to a big building, we are always surrounded by geometry in different shapes and sizes. The shape and surface to be taken up by such shapes need to be calculated so as to store them without any difficulty, or to meet the needs of the people using them. Such study or computation is given a special name in the field of mathematics.

What is Mensuration?

Whenever the dimensions of various geometrical shapes are computed and their area or capacity is measured in mathematics, it is termed mensuration. Mensuration also helps to calculate the dimensions of other shapes from their surface area and capacity.

Triangle

Such a 2-D shape is formed by joining three points, two of which are collinear is called a triangle. As the name suggests, such a shape has exactly three angles, the sum of whom is always 180°. ‘Tri’ suggests three vertices, sides, angles of this enclosed shape. It is geometrically expressed as ‘‘. In the figure below, ABC is a triangle with points B and C being collinear. Clearly, ABC has three sides AB, BC, and CA, and three angles ∠ABC, ∠BCA, and ∠CAB.

Types of Triangles

Four main types of triangles are as follows:

  • Scalene Triangle: Such a triangle that has angles that vary in measure and all three sides varying in length is called a scalene triangle.
  • Isosceles Triangle: Such a triangle that has two sides that are equal in length is called an isosceles triangle.
  • Equilateral Triangle: Such a triangle whose all sides are equal in length and all angles are equal in measure as well is called an equilateral triangle.
  • Right Triangle: Such a triangle that has an angle that measures 90° is called a right triangle. Such a triangle has three sides, out of which the hypotenuse is the longest and is opposite the 90 degrees angle.

Equilateral Triangle

Such a triangle whose all sides are equal in length and all angles are equal in measure as well is called an equilateral triangle. Since all the angles of a triangle total 180 degrees, hence each angle of an equilateral triangle shall measure 60°. In the figure below ◮ABC is an equilateral triangle whose three sides are equal in length, i.e., Ab = BC = CA, and each angle measures 60°, i.e., ∠ABC = ∠BCA = ∠CAB = 60°.

The perimeter of an equilateral triangle

The perimeter of a geometrical shape is defined as the sum of the lengths of all sides of the said shape. Hence, in order to calculate the perimeter of any triangle, be it scalene or isosceles or equilateral, one needs to know the lengths of sides of the said triangle to add them up to get the desired result. 

Since an equilateral triangle is the one whose all sides are equal in length, the sum of all the three (equal) sides would result in the sum being three times the length of an individual side of the given triangle. Hence, if the length of one side of an equilateral triangle is ‘a’ units:

Perimeter = a + a + a = 3a units, thrice the length of one side.

Area of an equilateral triangle

The area of an equilateral triangle, whose side measures ‘a’ units = \frac{\sqrt{3}}{4}(a)^2

How to find the length of one side of an equilateral triangle?

When the perimeter is given: Clearly the perimeter of an equilateral triangle is thrice the length of its side, dividing the perimeter by 3 would yield the length of the side of such a triangle. Hence the length of side of an equilateral triangle is one- third of the perimeter of the triangle.

Example: If the perimeter of an equilateral triangle is given to be 105 sq. units, find the length of its side.

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 105

⇒ L = 35 units

Sample Questions

Question 1. If the perimeter of an equilateral triangle is 15 units, find the length of its sides.

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 15

⇒ L = 5 units

Question 2. If the perimeter of an equilateral triangle is 75 units, find the length of its sides. 

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 75

⇒ L = 25 units

Question 3. If the perimeter of an equilateral triangle is 150 units, find the length of its sides.

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 150

⇒ L = 50 units

Question 4. If the perimeter of an equilateral triangle is 6 units, find the length of its sides.

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 6

⇒ L = 2 units

Question 5. If the perimeter of an equilateral triangle is 99 units, find the length of its sides.

Solution:

Side of an equilateral triangle = 1/3rd of its perimeter

⇒ L = 1/3 × 99

⇒ L = 33 units

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