Volume of a Cube
Volume of a cube is defined as the total number of cubic units occupied by the cube completely. A cube is a three-dimensional solid figure, having 6 square faces. Volume is nothing but the total space occupied by an object. An object with a larger volume would occupy more space. Let us understand the volume of a cube in detail along with the formula and solved examples in the following sections. Also, learn about the Surface area of the cube here.
What is the Volume of a Cube?
Volume of a cube is defined as the total capacity of the cube it is the total amount of liquid a cube can hold. The volume of a cube is measured in cubic units such as cm3, m3, etc.
A cube is a solid 3-D figure, with 6 square faces. All the faces of a cube are square hence it has all dimensions equal
Let the length, width, and height of a cube be ‘a’, then;
Volume of cube = a × a × a
Volume of Cube = a3
All corners of a cube meet at an angle of 90° degrees. The figure below shows a cube, where l is the length, b is the width, h is the height, and l = b = h. The length, width, and height represent the edges of the cube and when three edges meet at a point, it is called a vertex.
Volume of Cube Formula
Volume of a cube is defined as the total number of cube units that the cube occupies completely. A cube is a three-dimensional shape with six faces, twelve edges, and eight vertices. Therefore, the volume of a cube is the space surrounded by its six faces. Volume of the cube is calculated using two formulas which are discussed below:
Volume of Cube If Edge-Length is Given
The formula to calculate the volume of a cube when the side (Let a) of the cube is given
Volume of the cube = a × a × a
= a3
Thus, when the edge length is known volume of the cube can easily be found
Example: Find the volume of a cube with a side of 5 cm
Solution:
Given, edge length( a) =5 cm
Volume = 53 = 5 x 5 x 5 = 125 cm3
Volume of a Cube If Diagonal is Given
The formula to calculate the volume of a cube when the diagonal of the cube is given
Volume of the cube = [√3 × (d)3] / 9
where d is the diagonal of cube
Derivation of Volume of a Cube
The volume of any object is the space occupied by that solid in the 3-D plane. In a cube all the sides i.e. length, breadth, and height are equal (l = b = h). The formula for the volume of a cube is derived as follows:
A cube can be considered as the layers of squares which are stacked on top of one another. Thus, for the base of a square shape, the area is length multiplied by its breadth.
In square length and breadth are equal, thus the area will be “a2“.
A cube is made by adding multiple layers of square sheets on top of one other until the height becomes “a” unit. Thus, the height of the cube is “a”.
Now the volume of any regular figure is the base area multiplied by the height.
Thus,
Volume of Cube = a2 × a = a3 units3
How to Find Volume of a Cube?
Two methods by which the volume of a cube can be found are
- Using Edge-length
- Using Diagonal
Volume of a Cube is calculated using the steps discussed below:
Step 1: Note the dimension of the cube.
Step 2: Use the formula V = a3 where a is the length of the side of a cube, OR V = [√3 × (d)3] / 9, where d is the diagonal of the cube accordingly
Step 3: This is the required volume and it is measured in (unit)3
Solved Examples on Volume of Cube
Example 1: If the volume of a cube is 216 cm3, what is the dimension of the cube?
Solution:
Given, the volume of a cube, V = 216 cm3.
Volume of cube = (side)3 = (216)= (6)3.
Therefore, the side of cube is 6 cm.
Example 2: How many 3 cm × 3 cm × 3 cm cube boxes can fit in a large 15 cm cube box?
Solution:
Volume of each box = (3 × 3 × 3) cm3 = 27 cm3.
Volume of large cube box = (15 × 15 × 15) cm3 = 3375 cm3.
Number of boxes = Volume of large cube / Volume of small cube
= 3375cm3 / 27cm3
= 125 boxes.
Example 3: The volume of a cubic hard disk is 0.5 dm3. What are the dimensions of the disk?
Solution:
Since, the Volume of a cube = a3
0.5 = a3
a = 3√0.5
= 0.794 dm
Example 4: Calculate the volume of a cube with a diagonal of 3 inches.
Solution:
Given: Diagonal = 9 inch.
Cube Volume = [√3×(Diagonal)3]/9
Volume = √3×[(3)3/9]
= 3 × √ 3 = 3 × 1.732
= 5.196 inches3
Example 5: Find the edge of a cube whose volume is 1000 cm3
Solution:
Volume = 1000 cm3l
Volume = a3 where, a is edge
1000 = 103 = a3
a (edge) = 10 cm
Example 6: Find the volume of a cube of side 0.01 cm
Solution:
Given, Edge (a) = 0.01 cm
volume = a3
volume = (0.01)3 = 0.000001 cm3
FAQs on Volume of a Cube
Question 1: What is the volume of a cube?
Answer:
Volume of a cube is defined as the total capacity of a cube. It is the total amount of liquid a cube can hold.
Question 2: What is the unit of volume?
Answer:
The unit of volume is given by cubic units i.e. volume is always measured in m3, cm3, etc. It is generally measured in liters.
Question 3: Write the formula for the volume of a cube.
Answer:
The formula for the volume of the cube is
Volume = a3,
where a is the edge length of a cube
Question 4: What is the volume of a cube if diagonals of the cube are given?
Answer:
The formula for volume of a cube when diagonals are given is
Volume = [√3 × (d)3] / 9
where d is the diagonal of a cube
Question 5: What is the unit of volume of a cube?
Answer:
The unit of volume of a cube is the cube, or (unit)3. Also, the SI unit for volume is the cubic meter (m3), which is the volume occupied by a cube with sides of 1 m. Other important units are cubic feet (ft3), cubic centimeters (cm3), cubic millimeters (mm3), cubic inches (in3), cubic yards (yd3), etc.
Question 6; What is the volume of a cube of side 2a?
Answer:
The formula for the volume of a cube is
Volume = (side)3
side = 2a
volume = (2a)3 = 8a3
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