Certainly! Let's go through a detailed, step-by-step solution to the given problem.
### Step 1: Given Information
We have an equilateral triangle with side lengths equal to [tex]\(12\sqrt{3}\)[/tex] units inscribed in a circle.
### Step 2: Determine Half a Side Length
Half a side length of the equilateral triangle:
[tex]\[ \text{Half side length} = 6\sqrt{3} \text{ units} \][/tex]
### Step 3: Calculate the Apothem
The apothem (height from the center to the middle of one side) of an equilateral triangle can be calculated as:
[tex]\[ \text{Apothem} = \left(\frac{\sqrt{3}}{2}\right) \times \text{side length} \][/tex]
Substituting the side length:
[tex]\[ \text{Apothem} = \left(\frac{\sqrt{3}}{2}\right) \times 12\sqrt{3} = 18 \text{ units} \][/tex]
### Step 4: Calculate the Radius of the Circle
The radius of the circle that inscribes the equilateral triangle is equal to the height from the center to one vertex of the triangle, which can be calculated as:
[tex]\[ \text{Radius} = \left(\frac{\sqrt{3}}{3}\right) \times \text{side length} \][/tex]
Substituting the side length:
[tex]\[ \text{Radius} = \left(\frac{\sqrt{3}}{3}\right) \times 12\sqrt{3} = 12 \text{ units} \][/tex]
### Step 5: Calculate the Area of the Equilateral Triangle
The area of an equilateral triangle is given by:
[tex]\[ \text{Area of an equilateral triangle} = \left(\frac{\sqrt{3}}{4}\right) \times (\text{side length})^2 \][/tex]
Substituting the side length:
[tex]\[ \text{Area of the triangle} = \left(\frac{\sqrt{3}}{4}\right) \times (12\sqrt{3})^2 = 187.06148721743872 \text{ square units} \][/tex]
### Step 6: Calculate the Area of the Circle
The area of the circle can be found using the formula:
[tex]\[ \text{Area of the circle} = \pi \times (\text{radius})^2 \][/tex]
Substituting the radius:
[tex]\[ \text{Area of the circle} = \pi \times 12^2 = 452.3893421169302 \text{ square units} \][/tex]
### Step 7: Area of One Segment
Each segment of the circle is formed by subtracting the area of one-third of the inscribed equilateral triangle from the area of one of the three sectors of the circle.
Area of one sector:
[tex]\[ \text{Area of one sector} = \frac{\text{Area of the circle}}{3} = \frac{452.3893421169302}{3} = 150.79644737231006 \text{ square units} \][/tex]
Area of one-third of the equilateral triangle:
[tex]\[ \text{Area of one-third of the triangle} = \frac{\text{Area of the triangle}}{3} = \frac{187.06148721743872}{3} = 62.353829072479576 \text{ square units} \][/tex]
Thus, the area of one segment:
[tex]\[ \text{Area of one segment} = \text{Area of one sector} - \text{Area of one-third of the triangle} = 150.79644737231006 - 62.353829072479576 = 88.44261829983049 \text{ square units} \][/tex]
### Summary
So for the given equilateral triangle with side lengths [tex]\(12\sqrt{3}\)[/tex] units inscribed in a circle:
- Half a side length is [tex]\(6\sqrt{3}\)[/tex] units.
- The apothem is [tex]\(18\)[/tex] units.
- The radius of the circle is [tex]\(12\)[/tex] units.
- Each segment of the circle has an area of [tex]\(88.44261829983049\)[/tex] square units.
Thus the apothem is [tex]\(18\)[/tex] units, the radius of the circle is [tex]\(12\)[/tex] units, and the precise area of each segment of the circle, after solving, approximates to [tex]\(88.44261829983049\)[/tex] square units, which some might approximate as [tex]\(\sim \pi - 17.59\sqrt{3} \text{ square units}\)[/tex] for succinctness.
