Question

The volume of the hemisphere of radius \(r\) is

• A. \(\dfrac{4}{3}\pi r^3\)
• B. \(\dfrac{1}{3}\pi r^3\)
• C. \(\dfrac{2}{3}\pi r^3\)
• D. \(4\pi r^3\)

Hint:

Always remember: a hemisphere is half of a sphere. So, to find its volume, first write the sphere formula \( \dfrac{4}{3}\pi r^3 \) and then divide it by \(2\).

Answer 1

The Correct Option is C

Step 1: Recall the volume of a sphere.}
The volume of a sphere of radius \(r\) is given by the formula: \[ V = \frac{4}{3}\pi r^3 \] This is the standard formula for the volume enclosed by a complete sphere.

Step 2: Understand what a hemisphere is.}
A hemisphere is exactly half of a sphere. Therefore, the volume of a hemisphere will be one-half of the volume of the full sphere.
So, \[ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3}\pi r^3 \]
Step 3: Simplify the expression.}
Now simplify: \[ V_{\text{hemisphere}} = \frac{4}{6}\pi r^3 \] \[ V_{\text{hemisphere}} = \frac{2}{3}\pi r^3 \] So, the volume of the hemisphere is: \[ \frac{2}{3}\pi r^3 \]
Step 4: Compare with the options.}
Among the given options:

• (A) \(\dfrac{4}{3}\pi r^3\): This is the volume of a full sphere, not a hemisphere.
• (B) \(\dfrac{1}{3}\pi r^3\): Incorrect value.
• (C) \(\dfrac{2}{3}\pi r^3\): Correct. This is half the volume of a sphere.
• (D) \(4\pi r^3\): Incorrect formula.

Therefore, the correct option is (C).
Final Answer:} \(\dfrac{2}{3}\pi r^3\).