1. Understanding a Hemisphere
A hemisphere is exactly half of a sphere. It has:
- a curved surface (half of the sphere’s curved surface)
- a flat circular base
If the radius of the hemisphere is \(r\), then the base is a circle with area \(\pi r^2\).
Hemisphere shapes are common in domes, bowls, lamps, and certain containers.
2. Curved Surface Area of Hemisphere
The curved surface area (CSA) is the area of only the outer curved part, excluding the flat circular base.
For a hemisphere of radius \(r\):
\( \text{CSA} = 2\pi r^2 \)
This is exactly half of the surface area of a full sphere (which is \(4\pi r^2\)).
2.1. Example
A hemisphere has radius \(r = 5\,\text{cm}\). Its curved surface area is:
\( \text{CSA} = 2\pi r^2 = 2 \times \pi \times 25 = 50\pi \)
Using \(\pi = 3.14\):
\( 50 \times 3.14 = 157\,\text{cm}^2 \)
3. Total Surface Area of Hemisphere
The total surface area (TSA) includes:
- the curved surface area
- the flat circular base
The base has area:
\( \pi r^2 \)
So the total surface area becomes:
\( \text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2 \)
3.1. Example
If a hemisphere has radius \(r = 9\,\text{cm}\):
\( \text{TSA} = 3\pi r^2 = 3\pi \times 81 = 243\pi \)
Using \(\pi = 3.14\):
\( 243 \times 3.14 = 763.02\,\text{cm}^2 \)
4. Real-Life Applications
Hemisphere surface area is often used in:
- Construction of domes
- Designing bowls or containers
- Heat radiation calculations
- Making lampshades
If only the curved part is required (like polishing a bowl), use CSA. If the entire structure is exposed (like a dome placed on a surface), use TSA.