HOW TO FIND SIDE LENGTH OF CUBE FROM DIAGONAL

Deriving the Length of the Diagonal

In a cube each face will be square,

a² + a² = y²

2a² = y²

y = √2a²

y = a√2

Then,

a² + y² = x²

a² + (a√2)² = x²

a² + a²(2) = x²

a² + 2a² = x²

3a² = x²

x = √3a²

x = a√3

So, length of the diagonal of a cube is a√3.

Here a is side length.

Problem 1 :

Find the value of x.

Solution :

Problem 2 :

Find the lateral surface area of a cube, if its diagonal is √6 cm.

Solution :

Given, diagonal of the cube = √6 cm

Diagonal of the cube = √3 a

√3 a = √6

a = √6/√3

a = √2 cm

Lateral surface area of cube = 4a²

= 4 × (√2)²

= 4 × 2

= 8 cm²

Problem 3 :

The side length of a cube is 5 meters. What is the length of its diagonal across one of the faces?

Solution :

Side length of cube (a) = 5 m

Length of the diagonal of a cube = a√3

length of the diagonal = 5√3

Problem 4 :

Find the length of the face diagonal of a cube when the side measures 9 units. Use the diagonal face formula of a cube.

Solution :

Length of one face diagonal of cube = a√2

Side length of the cube = 9 units

length of one face of diagonal = 9√2

Problem 5 :

The length of the diagonal of a cube that can be inscribed in a sphere of radius 7.5 cm is

Solution :

The cube is covered by the sphere. Radius of the sphere = 7.5 cm

Half of the side length of cube = 7.5 cm

Side length of cube = 7.5(2)

= 15 cm

Length of diagonal =  a√2 (where a is the side length of cube)

= 15√2

Problem 6 :

Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is 12√3 cm. Find the edges of the three cubes

Solution :

Side length of cubes are 3x, 4x and 5x.

Volume of three cubes = (3x)³ + (4x)³ + (5x)³

= 27x³ + 64x³ + 125x³

= 216x³

Length of diagonal of cube = √3(side length)

12 √3 = √3(side length)

Side length of large cube = 12 cm

216x³ = 12³

(6x)³ = 12³

6x = 12

x = 12/6

x = 2

Side lengths of three cubes :

3x ==> 3(2) ==> 6 cm

4x ==> 4(2) ==> 8 cm

5x ==> 5(2) ==> 10 cm

So, the side lengths of cube is 6 cm, 8 cm and 10 cm respectively.

Problem 7 :

Find the length of the longest rod that can be placed in a room 12 m long, 9 m broad and 8 m high.

Solution :

Length = 12 m, width = 9 m and height = 8 m

Length of diagonal of cuboid = √l² + w² + h²

= √12² + 9² + 8²

= √(144 + 81 + 64)

= √289

= 17

So, 17 m is the length of the longest rod that can be placed in the room.

Problem 8 :

Find the length of the longest pole that can be placed in a room 12 m long 3 m wide and 9 m high.

Solution :

Length = 12 m, width = 3 m and height = 9 m

Length of diagonal of cuboid = √l² + w² + h²

= √12² + 3² + 9²

= √(144 + 9 + 81)

= √234

= 15.29

So, the longest pole that can be placed is 15.29 m.

Problem 9 :

If the length of a diagonal of a cube is 7√3 cm each of its side is

a)  7 cm    b) 49 cm     c) 343 cm    d)  14 cm

Solution :

Length of diagonal = 7√3

Side length of cube = 7 cm

So, option a is correct.

Problem 10 :

The length of a diagonal of a cube whose side measures 3 cm is ______________ 

Solution :

Side length of cube = 3 cm

Length of diagonal = √3(side length)

= 3√3

So, the required side length of the diagonal is 3√3 cm.

Problem 11 :

The length of an edge of a hollow cube open at one face is √3 meter. what is the length of the largest pole that it can accommodate?

Solution :

Side length of cube = √3 m

Length of diagonal = √3(side length)

= √3√3

= 3 meters

So, the longest pole can be fixed in the cube is 3 meters.

Problem 12 :

The surface area pf cube is 600 cm². Find the length of its diagonal.

Solution :

Surface area of cube = 600 cm²

4a² = 600

a² = 600/4

a² = 150

a = 5√6

Length of diagonal = √3(side length)

= √3(5√6)

= 5√(3 x 3 x 2)

= 10√3