medium difficulty6 min

Conversion of one solid into another

When one solid is melted, recast, or reshaped into another solid without wastage, the volume of material remains the same.

In one line: Recast means same material, same volume.

Practice This Concept

Understand it

In conversion problems, the important idea is conservation of material. The shape changes, but the amount of material does not change if nothing is lost. So the volume of the old solid is set equal to the volume of the new solid, and then the missing dimension is found.

Example

If a metal sphere is recast into a cylinder without loss, we write volume of sphere = volume of cylinder and solve for the unknown radius or height.

Formula and rule

For conversion problems without wastage, volume of the original solid = volume of the new solid.

See it worked out

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

Worked example

If a sphere is converted into a cylinder with no loss, write 4/3 πr³ = πR²h and solve the required unknown from the equality of volumes.

Method you can reuse

1Write the volume formula of the old solid.
2Write the volume formula of the new solid.
3Set both volumes equal.
4Substitute known values.
5Solve for the unknown quantity.

Sketch it in your answer

Draw both solids separately and label the given measurements. Mark the step where the volume formula is written for each solid before equating them.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A solid sphere is melted and recast into a cylinder without any loss of material. What should be kept equal?

Think first, then show answer

The volumes should be kept equal, because the amount of material remains the same when a solid is recast without wastage.

Common wrong answer

A common wrong answer is to compare surface areas or to assume the dimensions stay the same. In recasting, volume is the constant quantity.

Don't mix this up

Students often compare surface areas instead of volumes, or they forget that the total material stays the same.

Exam tip

In recasting questions, write volume before and after as equal, then substitute the correct formulas.

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 new radius, height, or number of smaller solids when a solid is reshaped, melted, or recast without wastage.

1
Enough for 1 mark
When one solid is melted, recast, or reshaped into another solid without wastage, the volume of material remains the same.
2
Add for 2 marks
Example: If a metal sphere is recast into a cylinder without loss, we write volume of sphere = volume of cylinder and solve for the unknown radius or height.
3
Add for 3 marks
In conversion problems, the important idea is conservation of material.

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