# Cube – Definition, Shape & Formula

Cube is a solid or hollow three-dimensional form of a square that has six square faces, eight vertices, and twelve edges. Some examples of a cube that we see regularly are sugar cubes, ice cubes, Rubik’s cubes, etc. The length, breadth, and height of a cube are the same as each face of a cube is a square.Â

A cube is also known as an equilateral cuboid, a square parallelepiped, or a right rhombic hexahedron and is one of the five platonic solids.

In this article, we will discuss what is a cube, cube shape, formulas related to cubes, properties, and examples.

## What is a Cube?

Cube is a 3D solid shape with six square faces with all 12 sides of equal length. A cube can be considered a special cuboid where all the length, breadth, and height are equal. A cube is a special case of a square prism having six square faces, eight vertices, and twelve edges.

We see various shapes in our daily lives that resemble the cube. Such as a 3Ã—3 Rubik’s Cube a puzzle famous among many, is an example of a cube. The image added below shows a cube along with its faces, edges, and vertices.

### Cube Definition

A cube is a three-dimensional geometric shape characterized by six equal square faces, twelve equal edges, and eight vertices (corners). Each of its six faces is a square, and all the angles between the faces are right angles (90 degrees)

Studying, the above figure, we conclude that two faces of cube have a common boundary called edge of cube and there are twelve (12) edges of cube, similarly on observing figure closely we conclude that a cube has six (6) faces and eight (8) vertices.

## Cube Shape

Cube shape is one of the fundamental shapes of mathematics and is observed very often in our daily lives. We can assume the cube is a polyhedron with equal length, breadth, and height. It can be easily stacked on one over another without leaving any spaces. We say about a cube that,

• It has Twelve (12) Edges
• It has Six (6) Faces
• It has Eight (8) Vertices

We see various types of figures in our daily life that are shaped like cubes that include, boxes, ice cubes, sugar cubes, etc.

### Cube Examples

Various Cube Examples are shown in the image below:

Cube Examples

## Properties of Cube

A cube is a 3D figure with equal dimensions having various properties. Some of the properties of the cube are,

• All the faces of a cube are square-shaped. Hence the length, breadth, and height of a cube are equal.
• The angle between any two faces of a cube is a right angle, i.e., 90Â°.
• Each face of a cube meets the other four faces.
• Three edges and three faces of a cube meet at a vertex.
• Opposite edges of a cube are parallel to each other.
• Faces or planes of a cube opposite to each other are parallel.

## Faces, Edges and Vertices in Cube

There are Six Faces, Tweleve Edges and Eight Vertices in a Cube

Let’s learn them in detail

### Cube Faces

There are 6 faces in a cube and exach faces are have same length and breadth. Hence, a cube has square faces

### Cube Edges

There are 12 edges in a cube. Cube Edges mark the boundary of the surfaces of cube

### Cube Vertices

There are 8 vertices in a cube. Cube Vertices are coners or the point of intersection of two or more edges

### Euler’s Formula in Cube

Euler’s Formula gives the relation between Faces, Edges and Vertices of a Polyhedron. Let’s verify the same for a cube. According to Euler’s Formula we know that

F + V = E + 2

where,

• F is Number of Faces,
• V is Number of Vertices
• E is Number of Edges

In a cube, F = 6, V = 8 and E = 12. Putting this value in above expression we get

LHS = F + V = 8 + 6 = 14

RHS = E + 2 = 12 + 2 = 14

Hence, F+ V = E + 2

Thus, cube satisfies Euler’s Formula

## Net of Cube

A cube is a 3-D figure and a figure in 2-D that can be folded easily to form the cube is called the net of a cube. Thus, we can say that the two-dimensional form of a cube that can be folded to form a three-dimensional form is called a net of a cube.

There are various ways to unfold a cube, i.e. a cube can have various nets one of nets of cube is discussed in image below,

## Cube Formula

There are various formulas that are helpful to find various dimensions of cube, that include length of its diagonal, its surface area, its volume, etc. various cube formulas discussed in article are,

Now let’s learn about these formulas in detail.

## Diagonal of Cube

Diagonal of a cube is the line segment that joins the opposite vertices of the cube. A cube has two types of diagonals, i.e., a face diagonal and a main diagonal. A face diagonal is a line that joins the opposite vertices of the face of a cube and is equal to the square root of two times the length of the side of a cube. As the cube has six faces, it has a total of 12 face diagonals. The formula to calculate the face diagonal of the cube is,

Length of Face Diagonal of Cube = âˆš2a units

where, a is Length of Side of a Cube

While the main diagonal is the line segment that joins the opposite vertices, passing through the center of the cube, and is equal to the square root of three times the length of the side of a cube. A cube has a total of four main diagonals.

Length of Main Diagonal of Cube = âˆš3a unitsÂ

where, a is Length of Side of a Cube

Below image represents main diagonal and face diagonal of cube.

## Surface Area of Cube

Area of any object is space occupied by all the surfaces of that object. It can be defined as the total surface available for the painting. A cube has six faces and so its surface area is calculated by finding the area of the individual face and finding its sum.

There are two types of surface area associated with a cube that are mentioned below,

• Lateral Surface Area of Cube, also called LSA of Cube.
• Total Surface Area of Cube, also called TSA of Cube.

### Lateral Surface Area of Cube

Lateral Surface Area of a cube is the sum of the areas of all the faces of a cube, excluding its top and bottom. In simple words, the sum of all four side faces of a cube is the lateral surface area of a cube. It is measured in square units such as (units)2, m2, cm2, etc.

Formula for the lateral surface of a cube is

Lateral Surface Area of Cube = 4a2

where,Â a is Length of Side of a Cube

### Total Surface Area of Cube

Total Surface Area of a cube is the space occupied by it in three-dimensional space and is equal to the sum of the areas of all its sides. It is measured in square units such as (units)2, m2, cm2, etc.

Formula for the total surface of a cube is

Total Surface Area of Cube = 6a2

where,Â a is Length of Side of a Cube

## Volume of Cube

Volume of a cube is the amount of space enclosed by the cube. It is usually measured in terms of cubic units. It is measured in cube units such as (units)3, m3, cm3, etc.

Formula for the volume of a cube is

Volume of a Cube = a3

where, a is Length of Side of a Cube

We can also calculate the volume of the cube if its diagonal is given, by using the formula,

Volume of Cube = (âˆš3d3)/9

where, d is Length of Main Diagonal of Cube

## Cube – Important Facts

The various important facts related to cube are mentioned below:

• All the faces of cube are equal in dimension and is square shaped
• The length, breadth and height of a cube are same
• We can say that cube is a cuboid with equal dimension in all the three directions
• Cube is one of the simplest polyhedrons
• The volume of cube is calculated by side Ã— side Ã— side
• Lateral Surface Area of Cube is calculated by 4 Ã— side2
• Total Surface Area of Cube is Calculated by 6 Ã— side2

## Cube Formula: Examples

Example 1: Find the total surface area of a cube if the length of its side is 8 units.

Solution:

Given,

• Length of side of Cube (a) = 8 units

We know that,

Total Surface Area of Cube (TSA) = 6a2

TSA = 6 Ã— (8)2Â

Â  Â  Â  Â  = 6 Ã— 64Â

Â  Â  Â  Â  = 384 square units.

Hence, the surface area of the cube = 384 square units.

Example 2: Find the volume of a cube if the length of its side is 5.5 inches.

Solution:

Given,

• Length of side of Cube (a) = 5.5 inches.

We Â know that,

Volume of Cube (V) = a3

V = (5.5)3Â

Â  Â = 166.375â€¬ cubic inches

Hence, the volume of the cube is 166.375â€¬ cubic inches.

Example 3: Find the length of the diagonal of a cube and its lateral surface area if the length of its side is 6 m.

Solution:

Given,

• Length of side of Cube (a) = 6 m

We know that,

Length of Diagonal of Cube(l) = âˆš3 a

l = âˆš3 Ã— 6Â

Â  = 6âˆš3 m

Lateral Surface Area of Cube (LSA) = 4a2

LSA = 4 Ã— (6)2Â

Â  Â  Â  Â  = 4 Ã— 36Â

Â  Â  Â  Â  = 144 m2

Hence, the length of the diagonal is 6âˆš3 m, and its lateral surface area is 144 square meters.

Example 4: Determine the length of the diagonal of the cube if the volume of the cube is 91.125 cm3.

Solution:

Given,

Volume of the cube (V) = 91.125 cm3

Let length of side of a cube be “s”

We have,Â

Volume of Cube = s3

s3 = 91.125

s = Â âˆ›(91.125)Â

Â  Â = 4.5 cm

Length of diagonal of a Cube(l) = âˆš3 s

l = âˆš3 Ã— 4.5 = 4.5âˆš3 cm.

Hence, the length of the diagonal is 4.5âˆš3 cm

Example 5: Determine the volume of the cube if its total surface area is 54 square units.

Solution:

Given,

Total Surface Area of Cube = 54 square units.

Let length of side of a cube be “a”

We have,

Surface Area of Cube (A) = 6a2

6a2 = 54

a2 = 54/6 = 9

a = âˆš9 = 3 units

We know that,

Volume of Cube (V) = a3

V = (3)3Â

Â  Â  = 27 cubic units

Hence, the volume of the cube = 27 cubic units

Example 6: Find the lateral surface area and the total surface area of a cube if the length of its diagonal is 5âˆš3 units.

Solution:

Given,

Length of Diagonal of Cube = 5âˆš3 units

Let length of side of a cube be “a”

We have,

Length of Diagonal of Cube = âˆš3 a

5âˆš3 = âˆš3 a

a = 5âˆš3/âˆš3 = 5 units

Lateral Surface Area of Cube (LSA) = 4a2

LSA = 4 Ã— (5)2Â

Â  Â  Â  Â  = 4 Ã— 25 = 100 square units.

Total Surface Area of Cube (TSA) = 6a2

TSA = 6 Ã— (5)2Â

Â  Â  Â  Â  = 6 Ã— 25Â

Â  Â  Â  Â  = 150 square units.

Hence, the lateral surface area of a cube is100 square units, and the total surface area is 150 square units.

Example 7: Find the length of the side of a cube if its lateral surface area is 196 square inches.

Solution:

Given,

Lateral Surface Area of Cube = 196 square inches

Let length of side of Cube be “a”

We have,

Lateral Surface Area of Cube (LSA) = 4a2

196 = 4a2

a2 = 196/4 = 49

a = âˆš49 = 7 inches

Hence, length of side of cube is 7 inches

## Cube Formula – Practice Questions

Try out following practice questions on Cube Formula

Q1. Find the Volume of a Cube with side 12 cm.

Q2. Find the edge of a cube whose volume is 216 cm3

Q3. Find the Diagonal of a Cube whose each side measures 3 cm

Q4. The volume of a cube is 343 cm3. Find its Lateral Surface Area and Total Surface Area

Q5. Find the amount of water that can be stored in a cubical tank of 1 m dimension

## Cube – Frequently Asked Questions

### What is a Cube?

A cube is a three-dimensional figure with equal length, breadth and height. It is formed by taking six square faces of equal sides together.

### What are Five Examples of Cube?

Five Examples of Cube are Ice Cube, Sugar Cube, Rubik’s Cube, Die of Ludo and Cubical Box

### How Many Faces, Edges and Vertices are in Cube?

There are six faces, eight vertices and twelve edges in a Cube

### What is the difference between a Cube and a Cuboid?

Cube and Cuboid both are three-dimensional figures and the basic difference between them is, that in a cube all the length, breadth and height are equal, whereas in cuboid the length breadth and height of the figure are different.

### What is the Formula to Calculate the Surface Area of a Cube?

Surface Area of a Cube is calculated using the formula,

• Lateral Surface Area of Cube (LSA) = 4a2Â
• Total Surface Area of Cube (TSA) = 6a2Â

### What is the Formula to Calculate the Volume of a Cube?

Volume of a Cube is calculated using the formula,

• Volume of Cube (V) = a3

### What are Cube Edges?

Cube Edges are the line segments joining two vertices of a cube

### How do You Find Vertex of a Cube?

You can find Vertex of Cube as the point of intersection of two edges of cube

### What is Cube Formula?

Cube Formula is given to calculate volume and surface area of a cube. Cube Formula is mentioned below:

• Volume of Cube = (Side)3
• TSA of Cube = 6 Ã— Side2
• LSA of Cube = 4 Ã— Side2

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