© 2026 LearnInShort
Melting and Recasting: One Rule, Many Shapes
Mensuration questions on solids look different every time: a sphere is melted into small balls, a cylinder is recast into cones, a metal ball is dropped into a tank of water. Yet almost all of them rest on one law: when a solid is melted, moulded or immersed, its volume is conserved. Material is neither created nor lost, only reshaped. So the working is always the same three steps: write the volume of the original solid, write the volume of the new solid, and equate the total volume before and after. The skill lies in knowing the formulas cold and in cancelling common factors such as π before multiplying. This short gives the volume and surface-area formulas for the cylinder, cone, sphere and hemisphere, then shows how the conservation rule solves melting, recasting and water-displacement questions in under a minute each.
1. A solid metal sphere of radius 6 cm is melted and recast into small spheres of radius 1 cm. How many small spheres are made?
2. What is the volume of a cylinder of radius 7 cm and height 10 cm? (Take π = 22/7)
3. A cone has base radius 6 cm and vertical height 8 cm. What is its slant height?
4. A cone and a cylinder have the same base radius and the same height. What is the ratio of the volume of the cone to the volume of the cylinder?
5. A solid cylinder of radius 3 cm and height 12 cm is melted and recast into cones of radius 3 cm and height 4 cm. How many cones are made?
6. What is the total surface area of a solid hemisphere of radius 7 cm? (Take π = 22/7)
7. If the radius of a sphere is doubled, its volume becomes how many times the original?
8. A metal sphere of radius 7 cm is dropped completely into a cylindrical vessel of radius 14 cm that holds water. By how much does the water level rise?