Question

Prove that the area of a equilateral triangle whose side is a

Answer 1

The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional surface. To recall, an equilateral triangle is a triangle whose all 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 which 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

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.

So,

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, drop a straight line from the top triangle vertex 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

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