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Total surface area of a cylinder

The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases.

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Student-friendly explanation

A closed cylinder has three exposed parts: one curved side and two circular ends. When a question asks for the outer or total surface area, all these exposed parts must be counted together. This is why the formula combines the curved part and the two bases into one expression.

How to write this in exams

  1. 1

    Start with the exact idea

    The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases.

  2. 2

    Then show how to use it

    1. Check that the cylinder is closed or all outer surfaces are needed. 2. Write TSA = 2πr(h+r). 3. Substitute the given r and h. 4. Simplify carefully. 5. Write the final answer with square units.

  3. 3

    Add one concrete example

    For a closed cylinder with radius 3 cm and height 8 cm, TSA = 2πr(h+r) = 2π × 3 × (8+3) = 66π cm², which is 207.43 cm² approximately.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 2πrh. That gives only the curved side and misses the two circular ends.

Definition

The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases.

Example

For a closed cylinder with radius 3 cm and height 8 cm, TSA = 2πr(h+r) = 2π × 3 × (8+3) = 66π cm², which is 207.43 cm² approximately.

Rule to remember

Total surface area of a cylinder = 2πr(h+r), which is the curved surface area 2πrh plus the two base areas 2πr².

Memory hook

Total means side plus both ends.

Examples and method

Worked example

For r = 3 cm and h = 8 cm, TSA = 2πr(h+r) = 2 × π × 3 × 11 = 66π cm².

Method to apply

1. Check that the cylinder is closed or all outer surfaces are needed. 2. Write TSA = 2πr(h+r). 3. Substitute the given r and h. 4. Simplify carefully. 5. Write the final answer with square units.

Diagram support

Draw a cylinder and clearly show the curved side and the two circular ends. Mark the exposed outer surfaces so the student can see why all three parts are included.

How CBSE asks it

Find the total outer area of a closed cylindrical can, drum, tank, or pipe end-to-end surface.

Avoid common mistakes

Common confusion

Students often forget one or both circular bases and write only the curved surface area.

Common wrong answer

A common wrong answer is 2πrh. That gives only the curved side and misses the two circular ends.

Exam tip

When the cylinder is closed or all outer surfaces are mentioned, use 2πr(h+r).

Quick check

A closed cylindrical water tank is painted on every outer surface. Which formula should be used?

Use 2πr(h+r), because the curved side and both circular bases are all part of the total outer surface.

Answer writing and exam use

1-mark answer

The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases.

2-mark answer

The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases. Total surface area of a cylinder = 2πr(h+r), which is the curved surface area 2πrh plus the two base areas 2πr². For a closed cylinder with radius 3 cm and height 8 cm, TSA = 2πr(h+r) = 2π × 3 × (8+3) = 66π cm², which is 207.43 cm² approximately.

3-mark answer

A closed cylinder has three exposed parts: one curved side and two circular ends. When a question asks for the outer or total surface area, all these exposed parts must be counted together. This is why the formula combines the curved part and the two bases into one expression. Total surface area of a cylinder = 2πr(h+r), which is the curved surface area 2πrh plus the two base areas 2πr². For r = 3 cm and h = 8 cm, TSA = 2πr(h+r) = 2 × π × 3 × 11 = 66π cm². Find the total outer area of a closed cylindrical can, drum, tank, or pipe end-to-end surface. A common wrong answer is 2πrh. That gives only the curved side and misses the two circular ends.
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