To find the perimeter of an equilateral triangle with an area of [tex]\( 9 \sqrt{3} \ \text{cm}^2 \)[/tex], we can follow these steps:
1. Recall the area formula of an equilateral triangle:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} a^2
\][/tex]
Here, [tex]\( a \)[/tex] is the side length of the equilateral triangle, and we know the area is [tex]\( 9 \sqrt{3} \ \text{cm}^2 \)[/tex].
2. Set up the equation with the given area:
[tex]\[
9 \sqrt{3} = \frac{\sqrt{3}}{4} a^2
\][/tex]
3. Solve for [tex]\( a^2 \)[/tex]:
Multiply both sides of the equation by 4:
[tex]\[
36 \sqrt{3} = \sqrt{3} a^2
\][/tex]
4. Divide both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[
36 = a^2
\][/tex]
5. Solve for [tex]\( a \)[/tex]:
[tex]\[
a = \sqrt{36} = 6 \ \text{cm}
\][/tex]
So, the side length of the equilateral triangle is 6 cm.
6. Find the perimeter of the equilateral triangle:
The perimeter [tex]\( P \)[/tex] is three times the side length [tex]\( a \)[/tex]:
[tex]\[
P = 3 \times a = 3 \times 6 = 18 \ \text{cm}
\][/tex]
Therefore, the perimeter of the equilateral triangle is [tex]\( 18 \ \text{cm} \)[/tex].
तसर्थ, समबाहु त्रिभुजको परिमिति [tex]\( 18 \ \text{cm} \)[/tex] छ।
