Area & Height of Equilateral Triangle| Proof of Formulae | Concept Clarification

Area and Height of Equilateral Triangle | Proof of Formulae | Understanding of Equilateral Triangle |Shortcut Tricks | Concept Clarification

In this video, we are going to learn the Area and Height of Equilateral triangle concept and formula. This video is being designed in a way to help all classes to learn about triangle . This video is enough to solve any type of problems because we believe in conceptual learning. Share this among your friends and learn Concept of Equilateral Triangle.

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Area Of Equilateral Triangle
The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle is a triangle in which all the sides are equal and the measure of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of its side is known.

Area of an Equilateral Triangle Formula
The formula for the area of an equilateral triangle is given as:

Area of Equilateral Triangle (A) = (√3/4)a2
Where a = length of sides

Learn more about isosceles triangles, equilateral triangles and scalene triangles here.

Derivation for Area of Equilateral Triangle
There are three methods to derive the formula for the area of equilateral triangles. They are:

Using basic triangle formula
Using rectangle construction
Using trigonometry
Deriving Area of Equilateral Triangle Using Basic Triangle Formula
Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.

Deriving Area Of Equilateral Triangle

Now,

Area of Triangle = ½ × base × height

Here, base = a, and height = h

Now, apply Pythagoras Theorem in the triangle.

a2 = h2 + (a/2)2

⇒ h2 = a2 – (a2/4)

⇒ h2 = (3a2)/4

Or, h = ½(√3a)

Now, put the value of “h” in the area of the triangle equation.

Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a)

Or, Area of Equilateral Triangle = ¼(√3a2)

Deriving Area of Equilateral Triangle Using Rectangle Construction
Consider an equilateral triangle having sides equal to “a”.

Equilateral Triangle
Equilateral Triangle

Now, draw a straight line from the top vertex of the triangle to the midpoint of the base of the triangle, thus, dividing the base into two equal halves.
Area Of Equilateral Triangle
Area Of Equilateral Triangle

Now cut along the straight line and move the other half of the triangle to form the rectangle.
How to find Area of Equilateral Triangle?
How to find Area of Equilateral Triangle?

Here, the length of the equilateral triangle is considered to be ‘a’ and the height as ‘h’

So the area of an equilateral triangle = Area of a rectangle = ½×a×h …………. (i)

Half of the rectangle is a right-angled triangle as it can be seen from the figure above.

Thereby, applying the Pythagoras Theorem:

⇒ a2 = h2 + (a/2)2

⇒ h2 = (3/4)a2

⇒ h = (√3/2)a ……………(ii)

Substituting the value of (ii) in (i), we have:

Area of an Equilateral Triangle

=(½)×a×(√3/2)a

=(√3/4)a2

Deriving Area of Equilateral Triangle Using Trigonometry
If two sides of a triangle are given, then the height can be calculated using trigonometric functions. Now, the height of a triangle ABC will be-

h = b. Sin C = c. Sin A = a. Sin B

Now, area of ABC = ½ × a × (b . sin C) = ½ × b × (c . sin A) = ½ × c (a . sin B)

Now, since it is an equilateral triangle, A = B = C = 60°

And a = b = c

Area = ½ × a × (a . Sin 60°) = ½ × a2 × Sin 60° = ½ × a2 × √3/2

So, Area of Equilateral Triangle = (√3/4)a2

Below is a brief recall about equilateral triangles:

What is an Equilateral Triangle?
There are mainly three types of triangles which are scalene triangles, equilateral triangles and isosceles triangles. An equilateral triangle has all the three sides equal and all angles equal to 60°. All the angles in an equilateral triangle are congruent.

Properties of Equilateral Triangle
An equilateral triangle is the one in which all three sides are equal. It is a special case of the isosceles triangle where the third side is also equal. In an equilateral triangle ABC, AB = BC = CA.

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