Question

The length of the diagonal of a cuboid is 13 cm. the volume is 144 cm cube and the total surface area 192 cm square
. Find the dimensions of the cuboid.​

Answer 1

The dimensions of the given cuboid are 4 cm, 12 cm, and 3 cm.

• Given,

Diagonal of a cuboid = 13 cm

We know, diagonal of a cuboid = √(l² + b² + h²)

where l is the length of the cuboid,

b is the breadth of the cuboid,

and h is the height of the cuboid.

• Therefore,

√(l² + b² + h²) = 13 cm

Or, l² + b² + h² = (13 cm)²

Or, l² + b² + h² = 169 cm²

• Given volume of the cuboid (V) =  144 cm³

Surface area if the cuboid (S.A.) = 192 cm²

• Volume of a cuboid is given by the formula :

V = l × b × h

=> 144 cm³ = l × b × h

• Surface area of a cuboid is given as:

S.A. = 2 (lb + bh + lh)

• Substituting l, b, and h in the algebric formula (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), we get,

(l + b + h)² = l² + b² + h² + 2 (lb + bh + lh)

Or, (l + b + h)² = 169 cm² + 192 cm²

Or, (l + b + h)² = 361 cm²

Or, l + b + h = √361 cm²

Or, l + b + h = 19 cm

• l × b × h = 144 cm³

The prime factors of 144 are 2 × 2 × 2 × 2 × 3 × 3

•The factors have to clubbed in a way that their sum results to 19, since l + b + h = 19 cm.

2 × 2 = 4, 2 × 2 × 3 = 12, 3

4 cm + 12 cm + 3 cm = 19 cm

• Therefore, the dimensions of the cuboid are 4 cm, 12 cm, and 3 cm.