C
CraftExam
medium difficulty5 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.

In one line: Cone holds one-third of a same base cylinder.

Practice This Concept

Understand it

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.

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.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

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

Method you can reuse

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
The cone volume is one-third of the cylinder volume when both have the same base radius and 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.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find the volume of conical containers, funnels, heaps, or solids formed like cones using the given radius and height.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    A cone narrows to a point, so it holds less space than a cylinder with the same circular base and height.

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