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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete 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.

Definition

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

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.

Rule to remember

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

Memory hook

Recast means same material, same volume.

Examples and method

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 to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

When one solid is melted, recast, or reshaped into another solid without wastage, the volume of material remains the same. For conversion problems without wastage, volume of the original solid = volume of the new solid. 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-mark answer

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. For conversion problems without wastage, volume of the original solid = volume of the new solid. 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. Find the new radius, height, or number of smaller solids when a solid is reshaped, melted, or recast without wastage. A common wrong answer is to compare surface areas or to assume the dimensions stay the same. In recasting, volume is the constant quantity.
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