Area Of Equilateral Triangle

An equilateral triangle is a special type of triangle with all three sides equal in length and all the angles equal in measurement, which is 60° each. These properties of an equilateral triangle sets it apart from other types of triangles. The area of an Equilateral Triangle is a fundamental property of this triangle, helping in solving various theoretical and practical questions. 

1.0Area Of Equilateral Triangle Formula 

The formula for the area of an equilateral triangle is an important tool in calculating the mathematical problems related to this triangle. The area of the equilateral triangle equation with side “a” is denoted with the letter “A” as: 

$A=\frac{\sqrt{3}}{4}\times a^2$

Here $\sqrt{3}$ is a constant, which is an integral part of the geometric relationship of the equilateral triangle, where $\sqrt{3}$=1.73. 

The Unit of the area of the equilateral triangle depends upon the unit of the side given. 

2.0Area Of Equilateral Triangle Formula Proof

The derivation of the area of an equilateral triangle can be derived with the help of using the basic triangle formula like this: 

To Prove: Area of Equilateral Triangle = $\frac{\sqrt{3}}{4}\times a^2$

Given: Sides of the triangle are: AB = AC = BC = a

Construction: Construct a line AD = h perpendicular to BC. 

Solution: In $△$ADB and $△$ADC

AB = AC (Given) 

AD = AD (Common)

∠ADB = ∠ADC = 90° (By construction) 

$△$ADB ⩭ $△$ADC (SAS)

$BD=DC(CPCT)$

$BD+DC=BC$

$BD+BD=BC$

$2BD=BC$

$BD=BC/2=a/2$

$\text{Area of a Triangle}=½\times base\times height$

$=½\times BC\times AD$…….(1)

To find AD, we need to apply Pythagoras theorem in the triangle $△$ADC

AC² = DC² + AD² 

$a^2=(\frac{a}{2}{)}^{2}2+A{D}^2$

$A{D}^2=a^2-\frac{a^2}{4}=\frac{4a^2-a^2}{4}=\frac{3a^2}{4}$

$AD=\frac{\sqrt{3}}{2}a$

Now, putting the value of AD in Equation (1)

$\text{Area of a Triangle}=\frac{1}{2}\times a\times \frac{\sqrt{3}}{2}a$

$\text{Area of a Triangle}=\frac{\sqrt{3}}{4}{a}^2$

3.0Properties Of An Equilateral Triangle

Several important properties related to an equilateral triangle include:

• All three sides of an equilateral triangle have the same length.  
• The measure of each angle of an equilateral triangle is equal to 60°, and due to this property, the equilateral triangle is also known as an equiangular triangle. 
• The perimeter of any given equilateral triangle is 3 times its side. i.e., Perimeter = 3a 
• In an equilateral triangle, the perpendicular bisector, median and altitude are all the same things.
• The centroid, circumcenter, incenter, and orthocenter are all at the same point, known as the centre of the triangle.

4.0Solved Examples

Problem 1: An equilateral triangle has an area of $\sqrt{3}$cm². Find the length of the side of the triangle.

Solution: According to the question

Area of an equilateral triangle = $\sqrt{3}$ cm²

$\frac{\sqrt{3}}{4}{a}^2=\sqrt{3}$

$a^2=4$

$a=2cm$

Hence, the length of the side of the triangle is 2cm. 

Problem 2: Calculate the area of an equilateral triangle, given that the length of the side of the triangle is 12cm. 

Solution: According to the question 

Length of side a = 12cm

Area of an equilateral triangle = $\frac{\sqrt{3}}{4}{a}^2$

$=\frac{\sqrt{3}}{4}\times 12^2=36\sqrt{3}c{m}^2$

Problem 3: A construction company is designing a roof for a triangular gazebo. The roof is shaped like an equilateral triangle with a side length of 20 meters. The roof will have a decorative triangular window in the centre, which is also an equilateral triangle. The side length of the window is 8 meters. Calculate the area of the roof and the window. Also, Find the area of the roof excluding the window. 

Solution: According to the question: 

Let the Length of the side of the roof = A = 20m

The area of the triangular gazebo = $\frac{\sqrt{3}}{4}{A}^2$

$=\frac{\sqrt{3}}{4}\times 20^2$

$=\frac{\sqrt{3}}{4}\times 400=100\sqrt{3}{m}^2$

Let the length of the window = $a=8m$

The area of the triangular window =$\frac{\sqrt{3}}{4}{a}^2$

$\frac{\sqrt{3}}{4}\times 8^2$

$=16\sqrt{3}{m}^2$

Area of roof excluding the window = The area of the triangular gazebo – The area of the triangular window

Area of roof excluding the window = $100\sqrt{3}-16\sqrt{3}=84\sqrt{3}{m}^2$