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. The volume of the cube is calculated by multiplying the length, breadth, and height of the cube. For a cube the length, breadth, and height are equal. Thus, the volume of a cube is just a side cube.
In this article, we will 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?
The 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 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 Side is Given
Formula to calculate the volume of a cube when the side (Let a) of the cube is given
Volume of 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
Volume = 5 × 5 × 5 = 125 cm3
Volume of a Cube If Diagonal is Given
Formula to calculate volume of a cube when diagonal of cube is given
Volume of Cube = [√3 × (d)3] / 9
where,
Volume of a Cube Equation
Equation which gives the volume of a cube is discussed below. Suppose a cube of edge length 'a' is taken then its volume is calculated using the formula.
Volume of Cube(V) = a × a × a = a3
For example, what is volume of cube if side length is 7 m?
Solution:
Side of cube =7 m
Volume of cube equation,
v = a3
Putting value of a in above equation we get,
v = (7)3
v = 343 m3
Thus, volume of cone is 343 m3
Derivation of Volume of a Cube
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). Formula for volume of a cube is derived as follows:
- A cube can be considered as layers of squares that are stacked on top of one another. Thus, for the base of a square shape, the area is length multiplied by its breadth.
- In a 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 volume of any regular figure is base area multiplied by height. Thus,
Volume of Cube = Base Area × Height = a2 × a = a3 units3
How to Find the 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 dimension of cube. Let side is represented by (a) and diagonal is represented as (d).
Step 2: Now use formula,
V = a3
OR
V = [√3 × (d)3] / 9
Step 3: Simplify above equation.
Step 4: Add unit3 to answer in step 3 to volume of cube.
As volume of a cube is a cubic function it increases drastically if we change the dimension of the cube. This can be understood by the following image.
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Surface Area of a Cube
Surface Area of Cube is the total area covered by all the faces of the cube. As a cube has six square faces of similar dimensions its volume is calculated by the formula,
Surface Area of Cube = 6a2
where,
Volume of a Cube and Cuboid
Cube is a three-dimensional figure with six faces and three dimensions length, breadth, and height but for a cube all the dimension length = breadth = height = a(say). Then its volume is given as,
Volume = a3
Cuboid is a three-dimensional figure with six faces and three dimensions length, breadth, and Height (l, b, and h) respectively then volume of cuboid is given by the formula:
Volume = l × b × h
Examples of Volume of a Cube from Everyday Life
Various examples which we come across in our daily life resembles cube and we are required to find their volume. Some of the common examples are,
- A cubical cardboard box is used to pack various objects.
- Some of the rooms we live in are shaped like cubes.
- An aquarium in the shape of a cube can hold water and the amount of water it can hold is calculated using the volume of cube formula, etc.
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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,
Volume of a cube, V = 216 cm3
Volume of cube = (side)3
V = (216) = (6)3
Therefore, 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(N) = (Volume of Large Cube) / (Volume of Small Cube)
N = 3375cm3 / 27cm3
N = 125 boxes
Thus, 125 boxes are required to fit in the large box.
Example 3: Volume of a cubic hard disk is 0.5 dm3. What are dimensions of disk?
Solution:
Volume of a Cube = a3
0.5 = a3
a = 3√0.5
a = 0.794 dm
Example 4: Calculate volume of a cube with a diagonal of 3 inches.
Solution:
Given,
Diagonal = 9 inch
Cube Volume = [√3 × (Diagonal)3] / 9
Volume(V) = √3×[(3)3/9]
V = √3 × 3
V = 1.732 × 3
V = 5.196 inches3
Example 5: Find edge of a cube whose volume is 1000 cm3
Solution:
Volume = 1000 cm3
Volume = a3
1000 = 103 = a3
a (edge) = 10 cm
Thus, edge of cube is 10 cm.
Example 6: Find volume of a cube of side 0.01 cm
Solution:
Given,
Edge (a) = 0.01 cm
Volume = a3
Volume(V) = (0.01)3
V = 0.000001 cm3
Thus, volume of cube is 10-6 cm3
Volume of a Cube Problems
Problem 1: A cube has a side length of 6 cm. Find its volume.
Problem 2: If the volume of a cube is 125 cubic units, what is the length of one side?
Problem 3: The volume of a cube is 512 cubic centimeters. Find the length of one side.
Problem 4: If the volume of a cube is 343 cubic meters, what is the length of one side?
Problem 5: A cube has a volume of 1000 cubic inches. Find the length of one side.
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