Question

The sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm and the length of its diagonal is `2sqrt(3)` cm. The total surface area of the cuboid is ______.

पर्याय

• 48 cm²

• 72 cm²

• 96 cm²

• 108 cm²

Answer 1

The sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm and the length of its diagonal is `2sqrt(3)` cm. The total surface area of the cuboid is `underline(bb(96  cm^2)`.

Explanation:

Let Length = l

Breadth = b

Height = h

Given that

Sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm

l + b + h = `6sqrt(3)` cm ......(1)

Also, given that

Length of its diagonal is `2sqrt(3)` cm

`sqrt(l^2 + b^2 + h^2) = 2sqrt(3)`  ......(2)

We need to find

Total surface area of the cuboid

Now, Total surface area of cuboid = 2(lb + bh + lh)

From (1)

l + b + h = `6sqrt(3)` cm

Squaring both sides

(l + b + h)² = `(6sqrt(3))^2`

l² + b² + h² + 2lb + 2bh + 2lh = `6^2 xx (sqrt(3))^2`

l² + b² + h² + 2lb + 2bh + 2lh = 36 × 3

l² + b² + h² + 2lb + 2bh + 2lh = 108

From (1)

`sqrt(l^2 + b^2 + h^2) = 2sqrt(3)`

Squaring both sides

`(sqrt(l^2 + b^2 + h^2))^2 = (2sqrt(3))^2`

l² + b² + h² = 2² × `(sqrt(3))^2`

l² + b² + h² = 4 × 3

l² + b² + h² = 12

12 + 2lb + 2bh + 2lh = 108

2lb + 2bh + 2lh = 108 – 12

2lb + 2bh + 2lh = 96

Total surface area = 96 cm²