medium difficulty6 min

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.

In one line: Hemisphere total means curved part plus base.

Practice This Concept

Understand it

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.

Example

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

Formula and rule

Curved surface area of a hemisphere = 2πr².

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

See it worked out

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

Worked example

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

Method you can reuse

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

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.

Don't mix this up

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

Exam tip

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

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

1
Enough for 1 mark
The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.
2
Add for 2 marks
Example: A hemisphere of radius 7 cm has total surface area 3πr² = 3π × 49 = 147π cm², which is 462 cm² when π = 22/7.
3
Add for 3 marks
A hemisphere is half of a sphere.

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