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Surface area of a hemisphere

The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.

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Student-friendly explanation

A hemisphere is half of a sphere. It has one curved half-spherical surface and one flat circular base. If the question asks for total surface area, both parts are counted, so the formula becomes 3πr². If only the curved part is asked, then 2πr² is used.

How to write this in exams

  1. 1

    Start with the exact idea

    The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.

  2. 2

    Then show how to use it

    1. Decide whether curved surface area or total surface area is required. 2. Use 2πr² or 3πr² accordingly. 3. Substitute the radius. 4. Simplify. 5. Write the answer in square units.

  3. 3

    Add one concrete example

    A hemisphere of radius 7 cm has total surface area 3πr² = 3π × 49 = 147π cm², which is 462 cm² when π = 22/7.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is 2πr² for every hemisphere question. That is correct only for the curved part, not for the total surface area.

Definition

The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.

Example

A hemisphere of radius 7 cm has total surface area 3πr² = 3π × 49 = 147π cm², which is 462 cm² when π = 22/7.

Rule to remember

Curved surface area of a hemisphere = 2πr². Total surface area of a hemisphere = 3πr², where r is the radius.

Memory hook

Hemisphere total means curved part plus base.

Examples and method

Worked example

For r = 7 cm, total surface area = 3πr² = 3 × π × 49 = 147π cm² = 462 cm².

Method to apply

1. Decide whether curved surface area or total surface area is required. 2. Use 2πr² or 3πr² accordingly. 3. Substitute the radius. 4. Simplify. 5. Write the answer in square units.

Diagram support

Draw a hemisphere and clearly show the circular base at the bottom. Shade the curved part and circle the base so students remember that total surface area includes both.

How CBSE asks it

Find the area of a hemispherical bowl, dome, cap, or half-ball when the whole outer surface or only the curved part is specified.

Avoid common mistakes

Common confusion

Students often use only 2πr² and forget that the circular base must also be included for total surface area.

Common wrong answer

A common wrong answer is 2πr² for every hemisphere question. That is correct only for the curved part, not for the total surface area.

Exam tip

Read the wording carefully: curved surface only means 2πr², but total surface area means 3πr².

Quick check

A bowl shaped like a hemisphere is open at the bottom. If the full outer surface is required, which formula is used?

Use 3πr² when the total surface area of a hemisphere is required, because it includes the curved part and the circular base.

Answer writing and exam use

1-mark answer

The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.

2-mark answer

The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for. Curved surface area of a hemisphere = 2πr². Total surface area of a hemisphere = 3πr², where r is the radius. A hemisphere of radius 7 cm has total surface area 3πr² = 3π × 49 = 147π cm², which is 462 cm² when π = 22/7.

3-mark answer

A hemisphere is half of a sphere. It has one curved half-spherical surface and one flat circular base. If the question asks for total surface area, both parts are counted, so the formula becomes 3πr². If only the curved part is asked, then 2πr² is used. Curved surface area of a hemisphere = 2πr². Total surface area of a hemisphere = 3πr², where r is the radius. For r = 7 cm, total surface area = 3πr² = 3 × π × 49 = 147π cm² = 462 cm². Find the area of a hemispherical bowl, dome, cap, or half-ball when the whole outer surface or only the curved part is specified. A common wrong answer is 2πr² for every hemisphere question. That is correct only for the curved part, not for the total surface area.
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