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The problem describes an equilateral triangle inscribed in a circle, where the side length of the triangle is 8 cm, and we are asked to find the total area of the shaded regions outside the triangle but within the circle.

To solve this, we can follow these steps:

### Step 1: Calculate the circumradius of the equilateral triangle
For an equilateral triangle with side length \( s \), the circumradius \( R \) (the radius of the circumscribing circle) is given by the formula:

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

For \( s = 8 \) cm:

\[
R = \frac{8}{\sqrt{3}} \approx 4.618 \, \text{cm}
\]

### Step 2: Calculate the area of the circle
The area \( A_{\text{circle}} \) of a circle is given by:

\[
A_{\text{circle}} = \pi R^2
\]

Substituting \( R = \frac{8}{\sqrt{3}} \):

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

### Step 3: Calculate the area of the equilateral triangle
The area \( A_{\text{triangle}} \) of an equilateral triangle with side length \( s \) is:

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

Substituting \( s = 8 \):

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

### Step 4: Calculate the area of the shaded region
The shaded region is the area of the circle minus the area of the triangle:

\[
A_{\text{shaded}} = A_{\text{circle}} - A_{\text{triangle}} = \frac{64\pi}{3} - 16\sqrt{3}
\]

### Conclusion
The total shaded area is:

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

This matches the last option provided in the choices.

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Would you like any further details on the steps? Here are five additional questions that expand on the solution:

1. How do you derive the formula for the circumradius of an equilateral triangle?
2. What is the relationship between the circumradius and the area of a triangle?
3. How do you calculate the area of a circle inscribed in an equilateral triangle?
4. Can you derive the formula for the area of an equilateral triangle from basic principles?
5. How would the problem change if the triangle were not equilateral?

**Tip:** Always double-check your formula usage when working with geometric figures involving circles and polygons to ensure precision in area calculations.

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

Learn how to find the total shaded area outside an equilateral triangle inscribed in a circle. The triangle has a side length of 8 cm, and we calculate the areas using geometry and area formulas. Suitable for high school students in Grades 10-12.

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