Boxed In
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
Age
11 to 14
Challenge level
Problem
Image
The diagram shows a rectangular box (a cuboid).
The areas of the faces are $3$, $12$ and $25$ square centimetres.
What is the volume of the box?
The areas of the faces of a cuboid are p, q and r.
What is the volume of the cuboid in terms of p, q and r?
Student Solutions
Catherine of Mount School, York and Joel of ACS Barker, Singapore realised that it is not necessary to calculate the lengths of the edges of the cuboid and they sent very similar solutions. This is Catherine's solution:
If the sides of the cuboid are x, y, and z and the areas of the rectangular faces are p, q and r then:
p = xy, q = yz and r = zx
It follows that pqr = (xy)(yz)(zx) = x 2 . y 2 . z 2 = (xyz) 2
So the volume = xyz = sqrt (pqr) = sqrt (3 x 12 x 25) = sqrt 900 = 30
Students from West Flegg Middle School, Norfolk and Madras College, St Andrew's and Russell Lower School, Ampthill also found the correct answer.