Technique

Recasting preserves volume

Mensuration • Recasting and Melting

Statement

When a solid is melted, rolled, drawn or poured into a new shape without loss, the volume of the new shape equals the volume of the old, so equating the two volume expressions determines the unknown dimension.

How to use it

1Write the volume of the old shape and of the new, set them equal, and solve for the single unknown length.
2Mass goes with volume for a uniform material, so a weight per unit volume scales the answer straight from the volume.
3For 'how many small pieces from one large solid', divide the large volume by the small one rather than equating.

Worked example

A metal cube of edge

6

cm is melted and recast as a cuboid

9

cm by

4

cm. What is its height?

Step-by-step

1
$6^{3} = 216$ cm $^{3}$
Why: The melted metal keeps its volume.
2
$9 \times 4 \times h = 216 \Rightarrow h = 6$ cm

Answer

6

Watch out

Volume is conserved; surface area is not, and neither is any single length. Where material is lost in the process, or a hollow shape is formed, the equality needs an explicit correction.

Builds on

Volume and wall area of a cuboid

Also known as

melting and recasting, volume is conserved, reshaping a solid, drawn into a wire, rolled into a bar

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2019 (II)

2017 (II)

2017 (I)

2016 (II)