Question

A cuboid has a total surface area of 96 cm². The sum of the squares of its length, breadth and height (in cm) is 48. The height of the cuboid is ________.

Answer 1

Let the length, breadth and height of the cuboid be l cm, b cm and h cm, respectively.

Total surface area of the cuboid = 96 cm² (Given)

⇒ 2(lb + bh + hl) = 96 cm² .....(1)

Also,

l² + b² + h² = 48 cm² (Given)

⇒ 2(l² + b² + h²) = 2 × 48 = 96 cm² .....(2)

Subtracting (1) from (2), we get

2(l² + b² + h²) − 2(lb + bh + hl) = 96 cm² − 96 cm² = 0

⇒ (l² − 2lb + b²) + (b² − 2bh + h²) + (h² − 2hl + l²) = 0

⇒ (l − b)² + (b − h)² + (h − l)² = 0

⇒ l − b = 0, b − h = 0, h − l = 0

⇒ l = b, b = h, h = l

⇒ l = b = h .....(3)

From (2) and (3), we get

h² + h² + h² = 48 cm²

⇒ 3h² = 48

⇒ h² = 16

⇒ h = 4 cm

Thus, the height of the cuboid is 4 cm.

A cuboid has a total surface area of 96 cm². The sum of the squares of its length, breadth and height (in cm) is 48. The height of the cuboid is ___4 cm___.