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Area of an Equilateral Triangle

The amount of space that an equilateral triangle covers in a 2-dimensional plane is called an Area Of an Equilateral Triangle. A triangle with all sides equal and all its angles measuring 60º is an equilateral triangle. The number of unit squares that can fit into is an area of any shape. “Unit” refers to one and a unit square is a square with a side of 1 unit. 

Area of an Equilateral Triangle

Finding an area of the triangle can be complicated and confusing but calculating the area of an equilateral triangle is quite an easy task. 

The general formula for the area of a triangle whose base and height are already given is: Area = 1/2 × base × height

Formula for calculating the area of an equilateral triangle is:

Area = √3/4 × (side)2 square units

If the given triangle ABC has an Area of ΔABC = (√3/4) × (side)2 square units, where AB = BC = CA = ‘a’ units (the length of equal sides of the triangle).

The formula for the area of the equilateral triangle can be written as:

Area of equilateral triangle ΔABC = (√3/4) × a2 square units

Area of an Equilateral Triangle Formula

If all the sides are equal and all the internal angles are 60°, only then it is said to be an Equilateral Triangle. If the length of one side of an equilateral triangle is known then the area of an equilateral triangle can be calculated. 

Area of an equilateral triangle formula:

Area of an equilateral triangle = (√3/4) × a2 square units

Here, a = Length of each side of an equilateral triangle.

The area of an Equilateral Triangle can be derived in many ways but three basic formulas to find the area of an equilateral triangle is:

  • Using the formula for general area of triangle
  • Using Heron’s Formula
  • Using Trigonometry

Deriving Equilateral Triangle’s Area Using Area of Triangle Formula:

If we know the length of each side and the height of the equilateral triangle then we can derive an area of the equilateral triangle using the general area of triangle formula. Thus we can calculate the height of an equilateral triangle in terms of the side length.

The formula for the area of an equilateral triangle is derived from the general formula of the area of the triangle which is  ½ × base × height. So Derivation for the formula of an equilateral triangle will be:

Area of triangle =  ½ × base × height

If we want to find the height of an equilateral triangle then in such case Pythagoras Theorem is used:

(hypotenuse2 = base2 + height2).

Let’s assume, base = a/2, height = h, and hypotenuse = a

Applying the Pythagoras theorem in the triangle.

a2 = h2 + (a/2)2

⇒ h2 = a2 – (a2/4)

⇒ h2 = (3a2)/4

Or, h = ½(√3a)

Next is, Putting the value of “h” in the area of the triangle equation.

Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a) , the base of the triangle is ‘a’ units.

Or, area of equilateral triangle = ¼(√3a2)

So, the formula of height comes as ½ × (√3 × side), and further, the area of the equilateral triangle becomes √3/4 × side2 square units.

Area of an equilateral triangle using Heron’s Formula:

When the lengths of the 3 sides of the triangle are known we can find the area of an equilateral triangle using Heron’s Formula. 

Consider the triangle ABC with sides a, b, and c. Heron’s formula to find the area of the triangle is:

Area = √s(s – a)(s – b)(s – c)

where,

s is the semi-perimeter which is given by:

s = (a + b + c)/2

For equilateral triangle: a = b = c.

s = (a + a + a)/2

s = 3a/2

Now, Area of equilateral triangle = √s(s−a)(s−a)(s−a)s(s−a)(s−a)(s−a)

= √3a2(3a2−a)(3a2−a)(3a2−a)3a2(3a2−a)(3a2−a)(3a2−a)

= √3a2(a2)(a2)(a2)3a2(a2)(a2)(a2)

= √3a2(a38)3a2(a38)

=√(3a416)(3a416)

Area of equilateral triangle = √3/4 × (side)2 square units.

Finding an Area of Equilateral Triangle With 2 Sides and an Included Angle (SAS):

To find the area of the triangle with 2 sides and included angles, use the sine trigonometric function to calculate the height of a triangle and use that value to find the area of the triangle. There are 3 variations to the same formula based on which sides and included angle are given.

Assuming a, b,c as the different sides of a triangle.

  • When sides ‘b’ and ‘c’ and included angle A are known, the area of the triangle is: 1/2 × bc × sin(A)
  • When sides ‘b’ and ‘a’ and included angle C are known, the area of the triangle is: 1/2 × ab × sin(C)
  • When sides ‘a’ and ‘c’ and included angle B are known, the area of the triangle is: 1/2 × ac × sin(B)

In an equilateral triangle, ∠A = ∠B = ∠C = 60°.Hence, sin A = sin B = sin C. Now, area of △ABC = 1/2 × b × c × sin(A) = 1/2 × a × b × sin(C) = 1/2 × a × c × sin(B).

Area = 1/2 × a × a × sin(C) = 1/2 × a2 × sin(60°) = 1/2 × a2 × √3/2

So, area of equilateral triangle = (√3/4)a2 square units.

Finding the Area of an Equilateral Triangle:

Given below are the steps that can be followed to find the area of an equilateral triangle using the side length:

  • Step 1: Write down the measure of the side length of the equilateral triangle.
  • Step 2: Then apply the formula to calculate the equilateral triangle’s area given as, A = (√3/4)a2, where, a is the measure of the side length of the equilateral triangle.
  • Step 3: Write the answer with the appropriate unit.

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