WORD PROBLEMS ON SURFACE AREA AND VOLUME OF CUBE

In Geometry, a Cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges.

Volume of cube = a³

Lateral surface area = 4a²

Total surface area = 6a²

To find side length of cube from the diagonal, we use the formula

Side length = a√3

Problem 1 :

Three cubes are joined end to end forming a cuboid. If side of a cube is 2 cm, find the dimensions of the cuboid thus obtained.

Solution :

Given side of a cube = 2 cm

After joining 3 cubes,

length of cuboid = 2 × 3 = 6 cm, 

Height = 2 cm and Breadth = 2 cm

Dimension of the cuboid is 6 cm x 2 cm x 2 cm.

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 :

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

Solution :

Diagonal of the single cube = 12√3 cm

√3 a = 12√3

a = 12 cm

Volume of the single cube = sum of the volumes of the metallic cubes

a³ = (3x)³ + (4x)³ + (5x)³

(12)³ = 27x³ + 64x³ + 125x³

1738 = 216x³

x³ = 1728/216

x³ = 8

x³ = 2³

x = 2

Now, the edge of the first cube = 3(2) = 6 cm

Edge of the second cube = 4(2) = 8 cm

Edge of the third cube = 5(2) = 10 cm

Therefore, the edges of the three cubes are 6 cm, 8 cm, and 10 cm.

Problem 4 :

Volume of a cube is 5832 m³. Find the cost of painting its total surface area at the rate of $3.50 per m².

Solution :

Volume of a cube is 5832 m³

a³ = 5832 m³

a = 5832

a = 18 m

Total surface area = 6 × a²

= 6 × 324 = 1944 m²

Cost of painting at 3.50 per m² = 1944 × 3.50

= 6804

Hence, the cost of painting is $6804

Problem 5 :

The cube has a surface area of 216 dm². Calculate:

a)   The area of one wall,

b)   Edge length,

c)   Cube volume.

Solution :

The cube has a surface area of 216 dm²

Problem 6 :

Find the side length of the cube whose surface area is 54 m²

Solution :

Surface area of cube = 54 m²

6a² = 54

a² = 54/6

a² = 9

a = 3 m

So, the required side length of the cube is 3 m.

Problem 7 :

A container of length 6 cm width 4 cm and height 9 cm is filled with orange juice. If the same amount of juices is to be stored in to a perfect cube shaped container what will be the size of its each side.

Solution :

Quantity of orange juice in the container = length x width x height

length = 6 cm, width = 4 cm and height = 9 cm

= 6 x 4 x 9

= 216 cm³

Amount of juice in the cuboid container = quantity of juice in the cube container

Let x be the side length of the cube.

216 = x³

x³ = 6³

x = 6 cm

So, side length of the cube is 6 cm.

Problem 8 :

The surface area of a cube is 1734 cm². Work out the volume of the cube.

Solution :

Surface area of a cube = 1734 cm²

6a² = 1734

a² = 1734/6

a² = 289

a = 17 cm

So, the side length of the cube is 17 cm.

Volume of cube = a³

= 17³

= 4913 cm³

Problem 9 :

The volume of a cuboid, whose length and breadth are equal, is 72 m³. If the cuboid’s height is 2 m, find its length and its surface area.

Solution :

Let l be the length, w be the width and h be the height of cuboid.

l = w and h = 2 m

Volume = 72

l x w x h = 72

l x l x 2 = 72

l² = 72/2

l² = 36

l = 6 m

Surface area of cuboid = 2(lw +wh + hl)

= 2(6 x 6 + 6 x 2 + 2 x 6)

= 2(36 + 12 + 12)

= 2(36 + 24)

= 2(60)

= 120 m²

Problem 10 :

A cube, of side length 10 cm, has the same volume as that of a cuboid of height 10 cm and width 8 cm. Find the length and the surface area of the cuboid.

Solution :

Side length of cube = 10 cm

length of cuboid = 10 cm

Width = 8 cm and height = ?

Volume of cube = volume of cuboid

10³ = 10 x 8 x h

1000 = 80 x h

h = 1000/80

h = 125

Surface area of cuboid = 2(lw + wh + hl)

= 2(10 x 8 + 8 x 125 + 125 x 10)

= 2(80 + 1000 + 1250) 

= 2(2330)

= 4660 cm²