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Surface Area of A Cube (Definition, Formula & Examples)

surface area of a cube

The surface area of a 3D shape (solid object) is a measure of the total area that the surface of the object occupies. The surface area of a cube is the total area covered by all six faces of the cube. The total surface area of a cube can be calculated if we calculate the area of the two bases and the area of the four lateral (side) faces. 

It’s important to know and understand the surface area as it helps you to know the amount of sheet of paper required to wrap a cube, paint the surfaces of the cube, etc. 

Let’s learn how to find the surface area of a cube and its uses.

Cube – A 3D Solid Shape

A cube(or regular hexahedron) is a 3D solid shape with six square faces and all the edges (or sides) of a cube are of the same length. The shape consists of six square faces, eight vertices, and twelve edges. The length, width(or breadth), and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length.

surface area of a cube

A cube has the following properties:

  • A cube has $12$ edges, $6$ faces, and $8$ vertices.
  • All the faces are shaped as a square hence the length, width(or breadth), and height are the same.
  • The angles between any two faces or surfaces are $90^{\circ}$.
  • The opposite planes or faces in a cube are parallel to each other.
  • The opposite edges in a cube are parallel to each other.
  • Each of the faces in a cube meets the other four faces.
  • Each of the vertices in a cube meets the three faces and three edges.

Difference Between Cube and Cuboid

Although cube and cuboid are similar $3D$ objects, there are few differences between these two. Following are the differences between a cube and a cuboid.

surface area of a cube
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Maths can be really interesting for kids

What is the Surface Area of a Cube?

The surface area of a cube is the sum of the area of the bases($2$ bases) and the area of lateral faces($4$ lateral faces) of the cube. Since all six faces of the cube are made up of squares of the same dimensions then the total surface area of the cube will be numerically equal to six times the area of one face. 

The surface area of a cube is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of a cube

  • Lateral Surface Area
  • Total Surface Area

Lateral Surface Area of a Cube

The lateral surface area of a cube refers to the total area covered by the side or lateral faces of a cube. There are four lateral faces in a cube, so to calculate LSA, we find the sum of the areas of these $4$ square faces.

surface area of a cube

If $a$ is the length of the edge (side) of a cube, then the area of one face (square) is $a \times a = a^{2}$ sq units.

Therefore, the lateral surface area (LSA) of a cube is $4 \times a^{2} = 4a^{2}$ sq units.

Formula for Lateral Surface Area = $4a^{2}$.

Total Surface Area of a Cube

The total surface area of a cube refers to the total area covered by all the faces of a cube. There are six square faces in a cube, so to calculate TSA, we find the sum of the areas of these $6$ faces.

surface area of a cube

If $a$ is the length of the edge (side) of a cube, then the area of one face (square) is $a \times a = a^{2}$ sq units.

Therefore, the total surface area (TSA) of a cube is $6 \times a^{2} = 6a^{2}$ sq units.

Formula for Total Surface Area = $6a^{2}$.

Examples

Ex 1: Find the area of the metal sheet required to make a cubical box of side length $4 cm$.

.Length of side of cubical box = $a = 4 cm$

Amount of metal sheet required to make a cubical box = Total Surface Area of a Cube = $6a^{2}$

$6a^{2} = 6 \times 4^{2} = 6 \times 16 = 96 cm^{2}$

Therefore, the area of the metal sheet required to make a cubical box of side length $4 cm$ is $96 cm^{2}$.

Ex 2: Find the ratio of the lateral surface area and total surface area of a cube.

Let the side of a cube be $a$ units

Lateral surface area = $4a^{2}$ sq units

Total surface area = $6a^{2}$ sq units

The ratio of LSA to TSA is $4a^{2} : 6a^{2} = \frac {4a^{2}}{6a^{2}} = \frac {4}{6} = \frac {2}{3} = 2 : 3$.

Ex 3: Find the length of the edge of the cube whose total surface area is $384 cm^{2}$.

TSA of cube = $384 cm^{2}$

$6a^{2} = 384 => a^{2} = \frac {384}{6} => a^{2} = 64 => a = 8 cm$

Length of the edge of a cube = $8 cm$

Maths in Real Life

Conclusion

The surface area of a 3D shape (solid object) is a measure of the total area that the surface of the object occupies. There are two types of surface areas in a cube, viz., lateral surface area calculated using the formula $4a^{2}$ and total surface area calculated using the formula $6a^{2}$, where $a$ is the length of each side of a cube.

Practice Problems

  1. If the length of the edge of a cube is $5 cm$, calculate its
    • lateral surface area
    • total surface area
  2. The surface area of a cube is 150 feet square. What is the length of the cube?
  3. If the total surface area of a cube is $96 cm^{2}$, find its lateral surface area.
  4. A cube of edge $2$ cm is divided into cubes of edge $1$ cm. 
    • How many cubes will be made?
    • Find the total surface area of the larger cube. 
    • Find the total surface area of the smaller cubes.
    • Which of the above two surface areas is larger?
  5. A solid cube of length $10 m$ is to be painted on its $6$ faces. If the painting rate is ₹$45$ per square metre, find the total cost of painting the cube.

Recommended Reading

FAQs

How to find the surface area of a cube?

Each face of the cube is a $2D$ plane figure which is a square. The area of a square is equal to $a^{2}$, where $a$ is the length of the edge(or side) of a cube. Since there are six faces of a cube, therefore the total surface area or surface area of a cube will be equal to the sum of areas of all its faces, i.e. $6a^{2}.

What is the formula for the surface area of a cube?

There are two types of surface areas in a cube.
a) The lateral surface area of a cube refers to the total area covered by the side or lateral faces of a cube. Its formula is $L = 4a^{2}$, where $a$ is the length of the edge(or side) of a cube.
b) The total surface area of a cube refers to the total area covered by all the faces of a cube. Its formula is $T = 6a^{2}$, where $a$ is the length of the edge(or side) of a cube.

What is the unit used to express the surface area of a cube?

The surface area of a cube is expressed in square units, for example using units like $cm^{2}$, $m^{2}$, $ft^{2}$, $in^{2}$ etc.

What is the total surface area of a cube?

The total surface area is the region occupied by all six faces of the cube in a three-dimensional space. The formula used to find the total surface area is $6a^{2}$, where $a$ is the edge(or side) length of the cube.

What is the lateral surface area of a cube?

The lateral surface area of a cube refers to the total area covered by the side or lateral faces of a cube in a three-dimensional space. The formula used to find the total surface area is $4a^{2}$, where $a$ is the edge(or side) length of the cube.

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