Question

An equilateral triangle with side 'a'. Find area of the triangle using Heron formula. If it's perimeter is 180 CM, what will be the area of the triangle.

Answer 1

Given :

• An equilateral triangle with side 'a'.
• Perimeter = 180 cm.

To Find :

• Area of the equilateral triangle using heron's formula.

Solution :

Since given in the question, the triangle is an equilateral triangle, we therefore know that each side of the triangle will be a cm.

Sum of the 3 sides i. e perimeter of the triangle is 180 cm.

So, from here we will find the length of each side.

$\longrightarrow$ $\sf{Perimeter\:=\:a+a+a}$

$\longrightarrow$ $\sf{180=3a}$

$\longrightarrow$ $\sf{\dfrac{180}{3}=a}$

$\longrightarrow$ $\sf{60=a}$

•°• Sides of the equilateral triangle, a = 60 cm.

Now, we have to find the area of the triangle with side 60 cm using the heron's formula.

Heron's formula :

$\large{\boxed{\sf{\purple{Area\:=\:\sqrt{s\:(s-a) (s-b) (s-c)}}}}}$

Where,

• s = Semi perimeter
• a = First side
• b = Second side
• c = Third side.

Since, the triangle is an equilateral triangle, we will substitute a instead of b and c.

Let's begin calculating the semi perimeter.

We know, semi perimeter (s) is ½ (half of) the perimeter.

$\sf{Semi\:perimeter\:(s)\:=\:\dfrac{Perimeter}{2}}$

$\longrightarrow$ $\sf{s=\dfrac{180}{2}}$

$\longrightarrow$ $\sf{s=90}$

$\large{\boxed{\sf{\blue{Semiperimeter\:(s)\:=\:90\:cm}}}}$

Now, substitute s = 90 and a = 60 in the heron's formula to find the area of the triangle.

$\sf{Area\:=\:\sqrt{90\:(90-60)(90-60)(90-60)}}$

$\longrightarrow$ $\sf{Area\:=\:\sqrt{90(30)(30)(30)}}$

$\longrightarrow$ $\sf{Area\:=\:\sqrt{90\:\times\:27000}}$

$\longrightarrow$ $\sf{Area\:=\:\sqrt{2430000}}$

$\longrightarrow$ $\sf{Area\:=\:1558.84}$

$\large{\boxed{\sf{\purple{Area\:of\:the\:triangle\:=\:1558.84\:cm^2}}}}$

Answer 2

$\bf{\underline{\underline{Given\: :}}}$

$\mathsf{\star\: \: Side\:of\:equilateral\: triangle = a}$

$\mathsf{\star\: \: Perimeter = 180\:cm}$

$\bf{\underline{\underline{To\:find\: :}}}$

$\mathsf{\star\: \: Area\:of \:the\:triangle}$

$\bf{\underline{\underline{Solution\: :}}}$

let us find out the side of the equilateral triangle

$\mathsf{Perimeter = 180\:cm}$

$\mathtt{\implies\: a + a + a = 180\:cm}$

$\mathtt{\implies\: 3a = 180\:cm}$

$\mathtt{\implies\: a = \frac{180}{3}\:cm}$

$\mathtt{\implies\: a = 60\:cm }$

Now, let us find out Semi perimeter (s)

$\large{\fbox{\mathtt{s = \frac{Perimeter}{2}}}}$

$\mathtt{\implies\: s = \Large{\frac{180}{2}\:}\large{cm}}$

$\mathtt{\implies\: s = 90\:cm}$

$\bf{\underline{\underline{Using\: Heron's \:formula\: :}}}$

$\fbox{\mathsf{Area = \sqrt{s(s-a)\:(s-b)\:(s-c)}}}$

$\mathtt{Where}$

• s = Semi perimeter
• a = first side
• b = second side
• c = third side

Since it is an equilateral triangle, we can express b and c as a

$\fbox{\mathsf{Area = \sqrt{s(s-a)\:(s-a)\:(s-a)}}}$

$\mathtt{\implies\: \sqrt{90(90 - 60) \: (90 - 60) \: (90 - 60) }}$

$\mathtt{\implies\: \sqrt{90 × 30 × 30 × 30 }}$

$\mathtt{\implies\: \sqrt{2700 × 900 }}$

$\mathtt{\implies\: \sqrt{2430000 }}$

$\fbox{\mathtt{\green{\implies\: 1558.84\: cm^{2} \:} (Answer) }}$