Question

If the diameter of a sphere is 'd' then its volume is

• A. \(\frac{1}{6}\pi d^3\)
• B. \(\frac{4}{3}\pi d^3\)
• C. \(\frac{1}{24}\pi d^3\)
• D. \(\frac{1}{3}\pi d^3\)

Answer 1

The Correct Option is A

To solve the problem, we need to determine the volume of a sphere given its diameter \( d \). Let us analyze this step by step.

1. Formula for the Volume of a Sphere:
The standard formula for the volume \( V \) of a sphere in terms of its radius \( r \) is:

$$ V = \frac{4}{3} \pi r^3 $$

2. Relating Diameter and Radius:
The diameter \( d \) of a sphere is twice its radius \( r \). Therefore:

$$ d = 2r $$

Solving for \( r \):

$$ r = \frac{d}{2} $$

3. Substituting \( r \) in the Volume Formula:
We substitute \( r = \frac{d}{2} \) into the volume formula:

$$ V = \frac{4}{3} \pi \left( \frac{d}{2} \right)^3 $$

Now, simplify the expression:

$$ V = \frac{4}{3} \pi \left( \frac{d^3}{8} \right) $$

$$ V = \frac{4}{3} \cdot \frac{1}{8} \pi d^3 $$

$$ V = \frac{4}{24} \pi d^3 $$

$$ V = \frac{1}{6} \pi d^3 $$

4. Conclusion:
The volume of the sphere in terms of its diameter \( d \) is \( \frac{1}{6} \pi d^3 \).

Final Answer:
The correct option is \( {\frac{1}{6} \pi d^3} \).