Example.1) The dimension of a cuboid are 15 cm, 12 cm, and 18 cm. Find the diagonal of the said cuboid.
Ans.) Here, l = 15 cm, b = 12 cm, h = 18 cm
As we all know the length of a diagonal of a cuboid = √(l² + b² + h²) = √(15² + 12² + 18²)
= √(225 + 144 + 324)
= √693 = 26.32 cm (approximately) (Ans.)
Example.2) The dimension of a cuboid are in the ratio 3 : 5 : 7 and its surface area is 9088 m². Find its (a) length, breadth, & height, (b) its volume.
Ans.) as per the given condition length : breadth : height = 3 : 5 : 7
l b h
Let, --------- = ---------- = --------- = y cm
3 5 7
So, l = 3y, b = 5y, and h = 7y
As we all know the formula for the surface area of the cuboid
= 2 (lb + bh + hl)
= 2{(3y X 5y) + (5y X 7y) + (7y X 3y)} = 2 (15y² + 35y² + 21y²) = 2 X 71y² = 142 y²
Given, 142 y² = 9088
Or, y² = 64
Or, y = √64 = √8²
Or, y = 8
So, Length = 3y = 3 X 8 = 24 cm
Breadth = 5y = 5 X 8 = 40 cm
Height = 7y = 7 X 8 = 56 cm ………………………………..(a) (Ans.)
The volume of the cuboid = l X b X h
= 24 cm X 40 cm X 56 cm
= 53760 cmᶟ ……………………..(b) (Ans.)
Example.3) The length of a diagonal of a cube is 11√3 cm. If length = 8 cm and breadth = 10 cm then find the height.
Ans.) as per the given condition, length = 8 cm and breadth = 10 cm.
let, height = H
so, as per the formula or rules we have = √ (l² + b² + h²)
so, √ (l² + b² + h²) = 11√3
so, {√ (l² + b² + h²)}² = (11√3)² [put square on both side]
so, l² + b² + h² = 363
so, 8² + 10² + H² = 363
so, H² = 363 – (64 + 100) = 363 – 164 = 199
so, H = √ 199 = 14.10 (approximately)
so the required height of the said cuboid is 14.10 cm (Ans.)
Example.4) If a side of a cuboid is 8 cm. find (a) its volume, (b) its surface area, (c) its lateral surface area, and (d) the length of its diagonal.
Ans.) the volume of the cube = (length of a side)ᶟ
= (8 cm)ᶟ = 512 cmᶟ ……………….(a)
Its surface area = 6 X (length of a side)²
= 6 X (8 cm)² = 6 X 64 cm² = 384 cm²…………….(b)
The lateral surface area of a cube = 4 X (length of a side)²
= 4 X (8 cm)² = 256 cm²………………..(c)
The length of its diagonal = √3 X length of a side
= √3 X 8 cm = 8√3 cm……………….(d) (Ans.)