Question

in area of eqalateral traingle is 9√3 cm² then the perimeter is ​

Answer 1

Given :-

• Area of equilateral triangle = 9√3 cm²

To find :-

• Perimeter of equilateral triangle.

According to the question,

$: \implies \sf{Area \: of \: equilateral \: \triangle = \dfrac{ \sqrt{3} }{4} {a}^{2} } \\ \\ : \implies \sf{9 \sqrt{3} = \frac{ \sqrt{3} }{4} {a}^{2} } \\ \\ :\implies \sf{9 \sqrt{3} \times 4 = \sqrt{3} {a}^{2} } \\ \\ : \implies \sf {36 \sqrt{3} = \sqrt{3} {a}^{2} } \\ \\ : \implies \sf{ \frac{36 \sqrt{3} }{ \sqrt{3} } = {a}^{2} } \\ \\ : \implies \sf{36 = {a}^{2} } \\ \\ : \implies \sf{ \sqrt{36} = a} \\ \\ : \implies { \boxed {\underline{\sf \red{6 = a}}}}$

• So, the side of equilateral triangle is 6 cm.

Now,

➞ Perimeter of equilateral triangle = 3a

➞ Perimeter of equilateral triangle = 3 × 6 cm

➞ Perimeter of equilateral triangle = 18 cm

• So, the perimeter of equilateral triangle is 18 cm.

____________________

Answer 2

Required Solution:

As per the provided question,we have:

• Area of the equilateral triangle = 9√3 cm²

We have to find:

• Perimeter

Step-by-step solution:

We know that,

• ${\underline {\boxed {\small {\bf \gray { Perimeter \: of \: ∆ = Sum \: of \: all \: sides } }}}}$

So,firstly we need to find sides:

As we know,

• ${\underline {\boxed {\small {\bf \gray { Area \: of \: an \: equilateral \: ∆ = \dfrac{\sqrt{3}}{4} {s}^{2} } }}}}$

Where,

• s = side
• Area = 9√3 cm² [Given]

Substitute the values to find side:

$\rm { \longmapsto 9 \sqrt{3} = \dfrac{ \sqrt{3} }{4} {(s)}^{2} }$

$\rm { \longmapsto 9 \sqrt{3} \times 4 = \sqrt{3} \times {(s)}^{2} }$

$\rm { \longmapsto 36 \sqrt{3} = \sqrt{3} \times {(s)}^{2} }$

$\rm { \longmapsto {(s)}^{2} = \dfrac{36 \cancel{\sqrt{3}} }{\cancel{\sqrt{3}} } }$

$\rm { \longmapsto {(s)}^{2} = 36}$

$\rm { \longmapsto s = \sqrt{36} }$

$\rm \purple { \longmapsto s = 6cm }$

Therefore, measure of the sides of the equilateral triangle is 6cm.

Now,

• ${\underline {\boxed {\small {\bf \gray { Perimeter \: of \: ∆ = Sum \: of \: all \: sides } }}}}$

$\rm { \longmapsto Perimeter = 6 + 6 + 6 }$

$\rm \purple { \longmapsto Perimeter = 18cm }$

Therefore, perimeter of the equilateral triangle is 18cm.

▬▬▬▬▬▬▬▬▬▬▬

Extra!

Some related formulas:

General triangle:

• Perimeter = a + b + c

Where, a,b,c are sides.

• Area = $\dfrac{1}{2}bh$

Equilateral triangle

• Perimeter = 3 × side
• Area = $\sf{ \dfrac{\sqrt{3}}{4} {s}^{2} }$

Right triangle

• Perimeter = Sum of all sides
• Area = $\dfrac{1}{2}bh$

Isosceles triangle

• Perimeter = 2a + d

Where, a = two equal sides

d = the one which is unequal side

• Area = $\sf{ \dfrac{1}{2} \times b \times \sqrt{ {a }^{2} - \dfrac{ {b}^{2} }{4} } }$

Where, a = two equal sides

b = base

▬▬▬▬▬▬▬▬▬▬▬