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Total surface area of a cone

The total surface area of a cone is the area of its curved surface plus the area of its circular base.

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Student-friendly explanation

A cone has one circular base and one curved sloping face. When the entire outer surface is required, both parts must be counted. The base area is added to the curved surface area, which gives the standard total surface area formula.

How to write this in exams

  1. 1

    Start with the exact idea

    The total surface area of a cone is the area of its curved surface plus the area of its circular base.

  2. 2

    Then show how to use it

    1. Confirm that the whole outer surface is required. 2. Write TSA = πr(l+r). 3. Substitute r and l. 4. Simplify carefully. 5. Present the answer in square units.

  3. 3

    Add one concrete example

    If a cone has radius 3 cm and slant height 5 cm, TSA = πr(l+r) = π × 3 × (5+3) = 24π cm².

  4. 4

    Avoid this incomplete answer

    A common wrong answer is πrl. That gives only the curved surface area and leaves out the circular base.

Definition

The total surface area of a cone is the area of its curved surface plus the area of its circular base.

Example

If a cone has radius 3 cm and slant height 5 cm, TSA = πr(l+r) = π × 3 × (5+3) = 24π cm².

Rule to remember

Total surface area of a cone = πr(l+r), where r is the radius and l is the slant height.

Memory hook

Cone total means curved side plus base.

Examples and method

Worked example

For r = 3 cm and l = 5 cm, TSA = πr(l+r) = π × 3 × 8 = 24π cm².

Method to apply

1. Confirm that the whole outer surface is required. 2. Write TSA = πr(l+r). 3. Substitute r and l. 4. Simplify carefully. 5. Present the answer in square units.

Diagram support

Draw the cone with the circular base clearly marked and the slant height on the side. Show that the outside area is the curved face plus the base circle.

How CBSE asks it

Find the outer covering of a conical cap, funnel, tent top, or solid cone including the base area.

Avoid common mistakes

Common confusion

Students often calculate only the curved surface area and forget to add the circular base.

Common wrong answer

A common wrong answer is πrl. That gives only the curved surface area and leaves out the circular base.

Exam tip

If the question says whole outer surface or complete covering, use πr(l+r).

Quick check

A conical tent is to be covered on the outside, including its base rim. Which formula should be used?

Use πr(l+r), because the total surface area of a cone includes both the curved side and the circular base.

Answer writing and exam use

1-mark answer

The total surface area of a cone is the area of its curved surface plus the area of its circular base.

2-mark answer

The total surface area of a cone is the area of its curved surface plus the area of its circular base. Total surface area of a cone = πr(l+r), where r is the radius and l is the slant height. If a cone has radius 3 cm and slant height 5 cm, TSA = πr(l+r) = π × 3 × (5+3) = 24π cm².

3-mark answer

A cone has one circular base and one curved sloping face. When the entire outer surface is required, both parts must be counted. The base area is added to the curved surface area, which gives the standard total surface area formula. Total surface area of a cone = πr(l+r), where r is the radius and l is the slant height. For r = 3 cm and l = 5 cm, TSA = πr(l+r) = π × 3 × 8 = 24π cm². Find the outer covering of a conical cap, funnel, tent top, or solid cone including the base area. A common wrong answer is πrl. That gives only the curved surface area and leaves out the circular base.
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