Find the perimeter, apothem, and area of a regular triangle with a radius of 6sqrt3.

How do I do this?

Did you make a sketch of an equilateral triangle?

draw in the "centre", and the "radius", and the height (the apothem)
I see a right-angled triangle with a hypotenuse of
6√3, and angles 30° , 60° and 90°
label the base as x and the height of the right-angled triangle as y, and the height of the whole triangle as h
sin30 = y/6√3
y = (6√3)(1/2) = 3√3
cos30 = x/6√3
x = (6√3)(√3/2) = 9
also:
tan 60 = h/9
h = 9√3

side of the triangle = 2x = 18
perimeter = 3(18) = 54
area = (1/2)(2x)(h) = xh = 9(9√3) = 81√3

To find the perimeter, apothem, and area of a regular triangle with a given radius, we need to understand a few key concepts.

First, let's define the terms:
1. Perimeter: The perimeter is the total length of all sides of the triangle.
2. Apothem: The apothem is the perpendicular distance from the center of the triangle to any side of the triangle.
3. Area: The area is the measure of the region enclosed by the triangle.

Now, let's break down the steps to find each value:

1. Perimeter:
Since it is a regular triangle, all three sides are equal in length. To find the perimeter, we need to find the length of one side and then multiply it by 3.

Knowing the radius, we can determine the length of one side using the formula for the circumradius (the distance from the center of the triangle to any vertex). The relation between the radius (r) and the circumradius (R) of a regular triangle is given by: R = r / cos(π/3).

Substituting the values, R = 6√3, we can solve for r:
6√3 = r / cos(π/3)

Next, we can determine the length of one side by using the formula for the side length (s) in terms of the radius (r):
s = 2 * r * sin(π/3)

Having obtained the side length (s), we can calculate the perimeter by multiplying it by 3.

2. Apothem:
The apothem of a regular triangle is equal to the distance from the center to any of its sides. In this case, the radius can be used as the apothem since it is also the distance from the center of the triangle to any vertex. The apothem is equal to the radius.

3. Area:
The area of a regular triangle can be determined using the formula: Area = (perimeter * apothem) / 2.

Now that we have the necessary steps, let's calculate the values:

1. Perimeter:
Substitute the value of R into the formula R = r / cos(π/3) to find r.
Calculate the side length using s = 2 * r * sin(π/3).
Multiply the side length by 3 to find the perimeter.

2. Apothem:
Use the given radius directly, since the apothem of a regular triangle is equal to the radius.

3. Area:
Calculate the area using the formula Area = (perimeter * apothem) / 2.

By following these steps, you can find the perimeter, apothem, and area of a regular triangle with a given radius.