Concept 12.3
Hollowing Out a Solid
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
If one solid is removed from another, the volume is the difference of the volumes, but the surface gains the inner surface that is now exposed.
The intuition
A wooden cylinder with a cone scooped out of its top: the wood left is the cylinder's volume minus the cone's.
The surface changes the other way. The cone's curved surface is now an inside wall you could paint, while only the small top circle is gone. Removing material can increase surface area.
The attention
Look at the new inside surface the hollow creates.
Volume: V(cyl) − V(cone) Surface: … + CSA of the cone, inside not − the cone's surface
Look hereNot here
- Look here
- The new inner surface. Scooping out a cone exposes its curved surface. That surface must be added.
- Not here
- Subtracting the cone's surface. Volume subtracts, so it's tempting to subtract surface too. But a hollow adds a wall.
- What it teaches
- Track what appears as well as what disappears.