Question

The volume of the right circular cone is _____.

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

Hint:

A cone is always “one-third of a cylinder” having the same radius and height: \[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} \] This makes the formula very easy to remember.

Answer 1

The Correct Option is D

Concept: A right circular cone is a three-dimensional solid having:
• a circular base,
• a fixed height,
• a single vertex called the apex. The volume of a cone measures the amount of space occupied inside it.

Step 1: Recall the formula for volume of a cylinder. A cylinder having radius \(r\) and height \(h\) has volume: \[ V_{\text{cylinder}} = \pi r^2 h \]

Step 2: Understand the relation between cone and cylinder. A cone having the same base radius and same height occupies exactly one-third the volume of such a cylinder. Therefore: \[ V_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}} \]

Step 3: Substitute cylinder volume. \[ V_{\text{cone}} = \frac{1}{3} \times \pi r^2 h \] \[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h \]

Step 4: Identify variables clearly.
• \(r\) = radius of circular base
• \(h\) = perpendicular height of cone
• \(\pi\) = constant approximately equal to \(3.14159\)

Step 5: Check the options carefully. Among all options, only: \[ \frac{1}{3}\pi r^2 h \] matches the standard cone volume formula. Final Answer: \[ \boxed{\frac{1}{3}\pi r^2 h} \]