C
CraftExam
medium difficulty5 min

Curved surface area of a cone

The curved surface area of a cone is the area of its sloping side, excluding the circular base.

In one line: Cone side means slant height, not vertical height.

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Understand it

A cone has one circular base and one curved sloping face. The sloping face is not measured by height; it is measured by the slant height. That is why the curved surface area uses radius and slant height, not the perpendicular height.

Example

For a cone with radius 5 cm and slant height 13 cm, CSA = πrl = π × 5 × 13 = 65π cm².

Formula and rule

Curved surface area of a cone = πrl, where r is the radius of the base and l is the slant height.

See it worked out

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

Worked example

For r = 5 cm and l = 13 cm, CSA = πrl = π × 5 × 13 = 65π cm².

Method you can reuse

  1. 1Identify radius and slant height.
  2. 2Check that only the curved side is needed.
  3. 3Use CSA = πrl.
  4. 4Substitute values correctly.
  5. 5Write the final answer in square units.

Sketch it in your answer

Draw a cone and label the radius on the base, the slant height along the sloping side, and the vertical height inside the cone. Shade only the curved face.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

An ice-cream cone is covered only on its sloping side. Which formula should be used for the covering?

Think first, then show answer
Use πrl, because only the curved sloping surface is covered and the slant height, not the vertical height, is required.

Common wrong answer

A common wrong answer is to use πrh.

Why it loses marks: A cone uses slant height, not perpendicular height, for curved surface area.

Don't mix this up

Students often use the vertical height instead of the slant height in the formula.

Exam tip

If only the side of the cone is covered, use πrl and remember that l means 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 area of the conical side of a tent, cap, funnel, or ice-cream cone from radius and slant height.
  1. 1

    Enough for 1 mark

    The curved surface area of a cone is the area of its sloping side, excluding the circular base.

  2. 2

    Add for 2 marks

    Example: For a cone with radius 5 cm and slant height 13 cm, CSA = πrl = π × 5 × 13 = 65π cm².

  3. 3

    Add for 3 marks

    The sloping face is not measured by height; it is measured by the slant height.

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