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The problem asks to find the total area of the three shaded regions outside an equilateral triangle inscribed in a circle.

### Step 1: Area of the Circle
The equilateral triangle has a side length of 8 cm. To find the radius of the circle in which this triangle is inscribed, we use the relation for the circumradius \( R \) of an equilateral triangle with side length \( s \):

\[
R = \frac{s}{\sqrt{3}}
\]

Substituting \( s = 8 \):

\[
R = \frac{8}{\sqrt{3}} \approx \frac{8\sqrt{3}}{3} \text{ cm}
\]

The area \( A_{\text{circle}} \) of the circle is:

\[
A_{\text{circle}} = \pi R^2 = \pi \left(\frac{8\sqrt{3}}{3}\right)^2 = \pi \times \frac{64 \times 3}{9} = \frac{192\pi}{9} \approx 21.33 \pi \, \text{cm}^2
\]

### Step 2: Area of the Equilateral Triangle
The area \( A_{\text{triangle}} \) of an equilateral triangle with side length \( s \) is given by:

\[
A_{\text{triangle}} = \frac{s^2 \sqrt{3}}{4}
\]

Substituting \( s = 8 \):

\[
A_{\text{triangle}} = \frac{8^2 \sqrt{3}}{4} = \frac{64 \sqrt{3}}{4} = 16 \sqrt{3} \, \text{cm}^2
\]

### Step 3: Area of the Shaded Region
The shaded regions are the areas of the circle outside the triangle, which can be found by subtracting the area of the triangle from the area of the circle:

\[
A_{\text{shaded}} = A_{\text{circle}} - A_{\text{triangle}} = \frac{192\pi}{9} - 16\sqrt{3} \, \text{cm}^2
\]

Thus, the total shaded area is:

\[
A_{\text{shaded}} = \frac{192\pi}{9} - 16\sqrt{3} \, \text{cm}^2
\]

### Step 4: Matching with Options
Let's simplify this and match it with the options provided:

The closest match to the formula is:

\[
\boxed{\frac{16}{3} \left(4\pi - 3\sqrt{3}\right)}
\]

Would you like further clarification or a breakdown of any specific part?

### Related Questions:
1. How would the shaded area change if the triangle had a different side length?
2. Can you derive the formula for the circumradius of an equilateral triangle?
3. What is the relationship between the side length of the triangle and the area of the circle?
4. How can this problem be generalized for polygons other than equilateral triangles?
5. What is the total perimeter of the shaded regions?

### Tip:
When solving geometry problems with inscribed shapes, understanding the relationships between the shapes' dimensions (like the circumradius or inradius) simplifies the problem significantly.

An equilateral triangle of side 8 cm is drawn so that its vertices lie on the circumference of a circle, as shown in the diagram. What is the total of the three areas shaded in the diagram, in cm²?

Solve the problem of finding the total shaded area in a circle where an equilateral triangle with side 8 cm is inscribed. Learn how to apply geometric formulas, including the area of a circle and an equilateral triangle, to calculate the shaded regions. Suitable for Grades 10-12 students.

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