# How do you find the perimeter of an equilateral with an altitude of #6sqrt3# cm?

Apply the Pythagorean theorem to solve for the side length and find that the perimeter is

Drawing it out, we obtain the following picture:

Our goal is to find the perimeter, that is,

Looking at the right triangle with the sides marked in the picture, we can apply the Pythagorean theorem to get

Substituting in our value for

Thus

So the perimeter is

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To find the perimeter of an equilateral triangle with an altitude of ( 6\sqrt{3} ) cm, you can use the formula:

[ \text{Perimeter} = 3 \times \text{side length} ]

Given that the altitude is ( 6\sqrt{3} ) cm, you can find the side length using the formula for the altitude of an equilateral triangle:

[ \text{altitude} = \frac{\sqrt{3}}{2} \times \text{side length} ]

Solve this equation for the side length:

[ 6\sqrt{3} = \frac{\sqrt{3}}{2} \times \text{side length} ]

[ \text{side length} = \frac{6\sqrt{3} \times 2}{\sqrt{3}} ]

[ \text{side length} = 12 ]

Now that you have the side length, you can find the perimeter:

[ \text{Perimeter} = 3 \times 12 ]

[ \text{Perimeter} = 36 \text{ cm} ]

So, the perimeter of the equilateral triangle is ( 36 ) cm.

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