Concept 12.4
Hemispheres: Is the Flat Face There?
Source: NCERT Mathematics, Class 106 minute lesson
Formal statement
A hemisphere of radius r has curved surface area 2πr² and total surface area 3πr² (curved surface plus the flat circular face). Its volume is (2/3)πr³.
The intuition
Half a sphere has half the sphere's curved surface, 2πr². But a solid hemisphere also has a flat face, a full circle of area πr².
So 'total' adds that circle: 3πr². Which one a question wants depends on whether the flat face is exposed. A bowl's inside uses 2πr²; a solid paperweight uses 3πr².
The attention
Look at the flat face: is it exposed or not?
Solid hemisphere: 2πr² + πr² = 3πr²
Look hereNot here
- Look here
- The flat circular face. The flat circle is the only difference between 2πr² and 3πr². Decide whether it is on the outside.
- Not here
- The curved surface. 2πr² alone fits a bowl, open on top. For a solid hemisphere it misses a whole circle.
- What it teaches
- When two formulas differ by one piece, the problem will tell you which piece is present.