To find the volume of a sphere, we can use the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( V \) is the volume and \( r \) is the radius of the sphere.
Given the diameter of the sphere is 8 cm, we can calculate the radius as follows:
\[ r = \frac{\text{diameter}}{2} = \frac{8 , \text{cm}}{2} = 4 , \text{cm} \]
Now we can substitute the radius into the volume formula:
\[ V = \frac{4}{3} \pi (4 , \text{cm})^3 \]
Calculating \( (4 , \text{cm})^3 \):
\[ (4 , \text{cm})^3 = 64 , \text{cm}^3 \]
Now substituting this back into the volume formula:
\[ V = \frac{4}{3} \pi (64 , \text{cm}^3) = \frac{256}{3} \pi , \text{cm}^3 \]
Next, we need to compute \( \frac{256}{3} \) and then multiply it by \( \pi \) (approximately 3.14):
\[ V \approx \frac{256}{3} \times 3.14 \approx 85.333 \times 3.14 \]
Calculating \( \frac{256 \times 3.14}{3} \):
\[ 256 \div 3 \approx 85.3333 \] \[ 85.3333 \times 3.14 \approx 267.999 \]
Rounding this to the nearest whole number gives us:
\[ 268 , \text{cm}^3 \]
Therefore, the volume of the sphere to the nearest whole number is:
\[ \boxed{268} , \text{cm}^3 \]
So the correct answer is C. 268 cm³.