What will happen to the volume of a cube if its edge is halved?

Cube is a three-dimensional solid object. It is also referred to as a regular hexahedron and is one of the five platonic solids. All the edges share at least a common edge with each other. The structure of a cube can be defined in terms of the fact that each of the faces is being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.

Properties of Cube 

• All the square faces are equal.
• All the sides are equal in length.
• Comprises of eight vertices and twelve edges.
• Length = Breadth = Height

Volume of Cube 

The total volume occupied by the cube is referred to as its volume. Since, a cube has all similar faces, therefore, the length of edges is also equivalent. Therefore, the length, width and height of the cube are of equal length. The volume of the cube is defined in terms of the cubic units that are occupied by the geometrical figure completely. The volume of the cube can be expressed in the form of centimeters³, meters³

The volume of the cube can be computed using the length of the edges of the cube. Let us assume the side of the cube to be a. Let us also assume V to be the volume of the cube. So, the volume of the cube formula is:

Volume of a Cube = Length × Width × Height

Since, all the sides are equal in cube.

Volume = a × a × a

Volume = a³

What will happen to the volume of a cube if its edge is halved?

Answer:

The formula used in solving this problem is, 

Volume of the Cube,  V = a³

Let us assume the side of the cube to be a.

Let us assume the volume of the cube to be ‘V’. 

Now, we know, 

The edge of the cube is reduced to half. 

This implies that,

a’ = a/2

Volume of the cube, V’ =  .

On simplifying, we get, 

The volume of the Cube when the edge is halved is  times the original volume.

Sample Questions 

Question 1. What will happen to the volume of the cube if the edge of the cube is doubled?

Solution:

Here we have to find the change in the volume of the cube when its edge is doubled

Assume a is the side of the cube

If the side of the cube is doubled

a = 2 × a = 2a

As we know that,

Volume of cube V =  a × a × a = a³

Volume of the cube of doubled side V’ =  2a × 2a × 2a

Volume of the cube of doubled side V’ =  8a³

Volume of the cube of doubled side V’ =  8 × V

Therefore,

We can see that if we double the edge of the cube its volume becomes 8 times.

Question 2. If the volume of a cube is 8000 m³, then find the side of the cube

Solution:

Here we have to find the length of the side of the cube,

Assume the side of the cube is ‘a’

As we know that,

Volume of cube  =  a × a × a = a³

In the question, we are given the volume of the cube as 8000 m³

8000 = a × a × a = a³

8000 = a³

a = 

a = 20 m

Therefore,

Side of the cube with a volume of 8000 m³ is 20 m.

Question 3. Find the increase in the volume of a cube of side 30 cm, if its sides are increased by 3 times?

Solution:

Here we have to find the percentage increase in the volume of 2 cubes 

Assume a is the side of the cube

As we know that

Volume of cube V = a × a × a = a³

V = 30 × 30 × 30 = 27000 cm³

Now assume the side of increased side cube be a’

a’ = 3 × a = 3a

As we know that

Volume of cube V’ = 3a × 3a × 3a

V’ = 27a³

V’ = 27 × V

Therefore,

We can see that if we increase the side of the cube by 3 times then its volume will be increased by 27 times.

Question 4. If two cubes of side 15 cm are joined to form a cuboid then find the volume of the cuboid formed by joining the two cubes?

Solution:

We can calculate the volume of the cuboid formed by joining two cubes by two methods

Method 1. By the formula of volume of a cuboid, in which the sides of cuboid will be; 

Length = 15 + 15 = 30 cm

breadth = 15 cm

height = 15 cm

Method 2. By finding the volume of single cube and then multiplying it by 2.

Find by using the formula of volume of cuboid,

Volume of Cuboid = Length × Breadth × Height

= l × b × h

= 30 × 15 × 15

= 6750 cm³

Find the volume by using volume of cube formula and then multiplying by 2.

Volume of the cube = a × a × a 

= a³

= 15³

= 3375 cm³

Now multiply by 2,

= 2 × 3375

= 6750 cm³

Question 5. Find out how much air can be filled in a cubical room of side 10 m?

Solution:

Here we have to find the air that can be filled in the cubical room

We are given that the side of the room is 10 m

As we know that

Volume of the cube = a × a × a 

Volume of the cube = a³

Volume of the cube = 10³

Volume of the cube = 1000 m³

Therefore,

1000 m³ air can be filled in a cubical room of size 10 m.