C
CraftExam
high importancemedium8 min

Volume of a cone

The volume of a cone is one-third of the volume of a cylinder having the same base radius and height.

Practice This Concept

Learn the concept

Student-friendly explanation

A cone narrows to a point, so it holds less space than a cylinder with the same circular base and height. In fact, three such cones fill one matching cylinder. That is why the cone volume formula has the factor one-third.

How to write this in exams

  1. 1

    Start with the exact idea

    The volume of a cone is one-third of the volume of a cylinder having the same base radius and height.

  2. 2

    Then show how to use it

    1. Identify the radius and perpendicular height. 2. Ignore slant height for volume. 3. Use 1/3 πr²h. 4. Substitute carefully. 5. Write the answer in cubic units.

  3. 3

    Add one concrete example

    For a cone with radius 7 cm and height 12 cm, volume = 1/3 πr²h = 1/3 × π × 49 × 12 = 196π cm³ = 616 cm³.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to use πrl or πr²h without the one-third factor. Both give the wrong space occupied by the cone.

Definition

The volume of a cone is one-third of the volume of a cylinder having the same base radius and height.

Example

For a cone with radius 7 cm and height 12 cm, volume = 1/3 πr²h = 1/3 × π × 49 × 12 = 196π cm³ = 616 cm³.

Rule to remember

Volume of a cone = 1/3 πr²h, where r is the radius of the base and h is the perpendicular height.

Memory hook

Cone holds one-third of a same base cylinder.

Examples and method

Worked example

For r = 7 cm and h = 12 cm, volume = 1/3 πr²h = 1/3 × π × 49 × 12 = 196π cm³ = 616 cm³.

Method to apply

1. Identify the radius and perpendicular height. 2. Ignore slant height for volume. 3. Use 1/3 πr²h. 4. Substitute carefully. 5. Write the answer in cubic units.

Diagram support

Draw the cone and show the perpendicular height from the top to the center of the base. Mark the radius on the base circle so the student does not confuse it with slant height.

How CBSE asks it

Find the volume of conical containers, funnels, heaps, or solids formed like cones using the given radius and height.

Avoid common mistakes

Common confusion

Students often forget the one-third factor or use slant height instead of perpendicular height.

Common wrong answer

A common wrong answer is to use πrl or πr²h without the one-third factor. Both give the wrong space occupied by the cone.

Exam tip

For cone volume, height must be perpendicular to the base, not the slant height.

Quick check

A conical cup and a matching cylinder have the same radius and height. How is the cone volume related to the cylinder volume?

The cone volume is one-third of the cylinder volume when both have the same base radius and height.

Answer writing and exam use

1-mark answer

The volume of a cone is one-third of the volume of a cylinder having the same base radius and height.

2-mark answer

The volume of a cone is one-third of the volume of a cylinder having the same base radius and height. Volume of a cone = 1/3 πr²h, where r is the radius of the base and h is the perpendicular height. For a cone with radius 7 cm and height 12 cm, volume = 1/3 πr²h = 1/3 × π × 49 × 12 = 196π cm³ = 616 cm³.

3-mark answer

A cone narrows to a point, so it holds less space than a cylinder with the same circular base and height. In fact, three such cones fill one matching cylinder. That is why the cone volume formula has the factor one-third. Volume of a cone = 1/3 πr²h, where r is the radius of the base and h is the perpendicular height. For r = 7 cm and h = 12 cm, volume = 1/3 πr²h = 1/3 × π × 49 × 12 = 196π cm³ = 616 cm³. Find the volume of conical containers, funnels, heaps, or solids formed like cones using the given radius and height. A common wrong answer is to use πrl or πr²h without the one-third factor. Both give the wrong space occupied by the cone.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept