Question

The areas of three adjacent faces of a cuboid are x, y, and z. If the volumes is V, prove that V² = xyz.

Answer 1

To prove : V² = xyz in the given cuboid.

• Let the length of the cuboid be denoted as l, breadth be denoted as b, and height be denoted as h.

• We know that,

Volume of a cuboid = length × breadth × height

 Volume of the given cuboid (V) = l × b × h

• Given,

Area of three adjacent sides of the cuboid = x, y, z

(The three adjacent sides have been shown in the image attached below)

• From the image,

Area of the longer lateral side = l × h

Area of the shorter lateral side = b × h

Area of the base = l × h

• According to the question,

l × h = x

b × h = y

l × h = z

 

• Now, V² = (l× b × h)²

Or, V² = l × b × h × l × b × h

Or, V² = (l × b) × (h × l) × (b × h)

Or, V² = x × z × y

Or, V² = xzy

Or, V² = xzy

Hence, proved.