Area Of An Equilateral Triangle - Formula, Derivation, Examples

Area of Equilateral Triangle

The area of an equilateral triangle is the amount of space that an equilateral triangle covers in a 2-dimensional plane. An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. The area of any shape is the number of unit squares that can fit into it. Here, 'unit' refers to one (1) and a unit square is a square with a side of 1 unit. Let us learn how to find the area of equilateral triangle using the equilateral triangle area formula with the help of solved examples.

1.	What is the Area of an Equilateral Triangle?
2.	Area of Equilateral Triangle Formula
3.	Area of Equilateral Triangle Proof
4.	How to Find Area of Equilateral Triangle?
5.	FAQs on Area of Equilateral Triangle

What is the Area of an Equilateral Triangle?

The area of an equilateral triangle is defined as the region covered within the three sides of the triangle and is expressed in square units. Some important units used to express the area of an equilateral triangle are in2, m2, cm2, yd2, etc. Let us understand the formula that is used to calculate the area of an equilateral triangle and its derivation in the following sections.

Area of an Equilateral Triangle Formula

The equilateral triangle area formula is used to calculate the space occupied between the sides of the equilateral triangle in a 2D plane. Calculating areas of any geometrical shape is a very important skill used by many people in their work. Finding the area of a scalene triangle or an isosceles triangle involves a few extra steps and calculations. However, finding the area of an equilateral triangle is comparatively easier.

The general formula for the area of a triangle whose base and height are known is given as:

Area = 1/2 × base × height

While the formula to calculate the area of an equilateral triangle is given as,

Area = √3/4 × (side)2

In the given triangle ABC, Area of ΔABC = (√3/4) × (side)2, where, AB = BC = CA = a units

Thus, the formula for the area of the above equilateral triangle can be written as:

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

Example: How to find the area of an equilateral triangle with one side of 4 units?

Solution:

Using the area of equilateral triangle formula: (√3/4) × a2,

we will substitute the values of the side length.

Therefore, the area of the equilateral triangle (√3/4) × 42 = 4√3 square units.

Area of Equilateral Triangle Proof

In an equilateral triangle, all the sides are equal and all the internal angles are 60°. So, an equilateral triangle’s area can be calculated if the length of one side is known. The formula to calculate the area of an equilateral triangle is given as,

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

where,

a = Length of each side of an equilateral triangle

The above formula to find the area of an equilateral triangle can be derived in the following ways:

• Using the general area of a triangle formula
• Using Heron's formula
• Using trigonometry

Deriving Equilateral Triangle's Area Using Area of Triangle Formula

The formula used to calculate the area of an equilateral triangle can be derived using the general area of the triangle formula. To do so, we require the length of each side and the height of the equilateral triangle. We will calculate the height of an equilateral triangle in terms of the side length.

The formula for the area of an equilateral triangle comes out from the general formula of the area of the triangle which is equal to ½ × base × height. The derivation for the formula of an equilateral triangle is given below.

Area of triangle = ½ × base × height

For finding the height of an equilateral triangle, we use the Pythagoras theorem (hypotenuse2 = base2 + height2).

Here, base = a/2, height = h, and hypotenuse = a (refer to the figure given above).

Now, apply the Pythagoras theorem in the triangle.

a2 = h2 + (a/2)2

⇒ h2 = a2 - (a2/4)

⇒ h2 = (3a2)/4

Or, h = ½(√3a)

Now, substitute this 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)

Therefore, the area of equilateral triangle = √3/4 × side2

Deriving Area of Equilateral Triangle Using Heron's Formula

Heron's formula is used to find the area of a triangle when the lengths of the 3 sides of the triangle are known. In mathematics, Heron's formula is named after Hero of Alexandria, who gives the area of any triangle when the lengths of all three sides are known. We do not use angles or other parameters for finding the area of a triangle using Heron's formula.

The following steps show the derivation of the formula for finding the area of a triangle:

Consider the triangle ABC with sides a, b, and c. Using the Heron's formula, we will find the area of the triangle is:

Area = √s(s - a)(s - b)(s - c)

where,

s is the semi-perimeter which is calculated as follows:

s = (a + b + c)/2

For an equilateral triangle, a = b = c

s = (a + a + a)/2

s = 3a/2

Now, Area of equilateral triangle = \(\sqrt {s(s - a)(s - a)(s - a)}\)

= \(\sqrt {\frac{{3a}}{2}(\frac{{3a}}{2} - a)(\frac{{3a}}{2} - a)(\frac{{3a}}{2} - a)}\)

= \(\sqrt {\frac{3a}{2}(\frac{{a}}{2})(\frac{{a}}{2})(\frac{{a}}{2})}\)

= \(\sqrt{\frac{3a}{2}(\frac{{a^3}}{8}})\)

=\(\sqrt{({\frac{{3a^4}}{16}})}\)

Therefore, Area of equilateral triangle = √3/4 × (side)2

Deriving Area of Equilateral Triangle With 2 Sides and Included Angle (SAS)

For finding the area of a triangle with 2 sides and the included angle, 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 three variations to the same formula based on which sides and included angle are given.

Consider a, b, and c are the different sides of a triangle.

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

In an equilateral triangle, ∠A = ∠B = ∠C = 60°. Therefore, 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).

For an equilateral triangle, a = b = c (refer to the figure given above).

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

How to Find the Area of Equilateral Triangle?

The following steps can be followed to find the area of an equilateral triangle using the side length:

• Step 1: Note the measure of the side length of the equilateral triangle.
• Step 2: 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: Express the answer with the appropriate unit.

Now, that we have learned the formula and method to calculate the area of the equilateral triangle, let us see a few solved examples to find the area of an equilateral triangle.