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Curved surface area of a cone

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to use πrh. That is incorrect because a cone uses slant height, not perpendicular height, for curved surface area.

Definition

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

Example

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

Rule to remember

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

Memory hook

Cone side means slant height, not vertical height.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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.

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

A common wrong answer is to use πrh. That is incorrect because a cone uses slant height, not perpendicular height, for curved surface area.

Exam tip

If only the side of the cone is covered, use πrl and remember that l means slant height.

Quick check

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

Use πrl, because only the curved sloping surface is covered and the slant height, not the vertical height, is required.

Answer writing and exam use

1-mark answer

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

2-mark answer

The curved surface area of a cone is the area of its sloping side, excluding the circular base. Curved surface area of a cone = πrl, where r is the radius of the base and l is the slant height. For a cone with radius 5 cm and slant height 13 cm, CSA = πrl = π × 5 × 13 = 65π cm².

3-mark answer

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. Curved surface area of a cone = πrl, where r is the radius of the base and l is the slant height. For r = 5 cm and l = 13 cm, CSA = πrl = π × 5 × 13 = 65π cm². Find the area of the conical side of a tent, cap, funnel, or ice-cream cone from radius and slant height. A common wrong answer is to use πrh. That is incorrect because a cone uses slant height, not perpendicular height, for curved surface area.
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