Question

a traffic signal board indicating School ahead is an equilateral triangle with side a find the area of signal board using heron's formula with the perimeter is 180 centimetres what will be the area of the signal board?​

Answer 1

The perimeter of a triangle is equal to the sum of its three sides it is denoted by 2S.

2s=(a+b+c)

s=(a+b+c)/2

Here ,s is called semi perimeter of a triangle.

 

The formula given by Heron about the area of a triangle is known as Heron's formula. According to this formula area of a triangle= √s (s-a) (s-b) (s-c)

Where a, b and c are three sides of a triangle and s is a semi perimeter.

 

This formula can be used for any triangle to calculate its area and it is very useful when it is not possible to find the height of the triangle easily . Heron's formula is  generally used for calculating area of scalene triangle.

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Solution:

Given, side of a signal whose shape is an equilateral triangle= a

Semi perimeter, s=a+a+a/2= 3a/2

Using heron’s formula,

Area of the signal board = √s (s-a) (s-b) (s-c)

= √(3a/2) (3a/2 – a) (3a/2 – a) (3a/2 – a)

= √3a/2 × a/2 × a/2 × a/2

= √3a⁴/16

= √3a²/4

Hence, area of signal board with side a by using Herons formula is= √3a²/4

 

Now,

Perimeter of an equilateral triangle= 3a

Perimeter of the traffic signal board = 180 cm  (given)

3a = 180 cm

a = 180/3= 60 cm

Now , area of signal board=√3a²/4

= √3/4 × 60 × 60

= 900√3 cm²

Hence , the area of the signal board when perimeter is 180 cm is 900√3

Hope this will help you....

Answer 2

Given :-

• Side of an equilateral triangle is a
• Perimeter is 180 cm

Need To Find :-

• The area of the board using Heron's Formula

Solution :-

$\small\underline{\pmb{\sf Heron's\: Formula \: is \: given \: by :-}}$

$\underline{\boxed{\sf\sqrt{s(s-a)(s-b)(s-c)}}}$

• We are given, perimeter of the triangle is 180 cm.Let's find it’s semi-perimeter.

$\sf :\implies Semi-\ Perimeter = \dfrac{180}{2}$

$\sf \green{:\implies Semi- \ Perimeter = 90}$

$\small\underline{\pmb{\sf According \: to \: the \: question :-}}$

$\sf:\implies a+a+a=180$

$\sf :\implies 3a=180$

$\sf :\implies a=\dfrac{180}{3}$

$\sf \pink{ :\implies a=60}$

$\small\underline{\pmb{\sf Using \: Heron's\: Formula :-}}$

$\begin{gathered}\;{\boxed{\sf{\purple{Area_{Signals\:board} = \sqrt{s(s - a)(s - b)(s - c)}}}}}\\ \\\end{gathered}$

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

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

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

$\sf :\implies Area =\sqrt{90(2700)}$

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

$\sf :\implies Area= \sqrt{243 \times 10^4}$

$\sf :\implies Area= 3\times 3 \times 100\sqrt{3}$

$\sf \pink{\sf{:\implies Area = 900\sqrt{3} \ cm^2}}$

$\therefore\:\underline{\textsf{Area of signal board is \textbf{900√3cm² }}}.$