Surface Area of Cube

The surface area of a cube is defined as the total area covered by all the sides of a cube. In geometry, a cube is a solid three-dimensional form of a square. A cube has six square faces, eight vertices, and twelve edges. A Rubik’s cube, sugar cubes, an ice cube, dice, etc., are some examples of cubes. The faces of a cube share a common boundary called the edge. The edge of a cube depicts the length, breadth, and height of a cube, which are connected at the vertex. Since the six faces of a cube are squares, the length, breadth, and height of a cube are equal. A cube is one of the five platonic solids and is also known as an equilateral cuboid, a square parallelepiped, or a right rhombic hexahedron. As all the sides of a cube are square-shaped, a cube is a particular case of a square prism.

What is the Surface Area of Cube?

The surface area of a cube is the sum of the areas of all sides. The region occupied by any shape is called the area. The total area covered by all six sides or faces of a cube is called the surface area of a cube. Hence, the total surface area of a cube is the sum of the areas of its six faces or sides. The total surface area of a cube is equal to six times the square length of the sides of a cube, i.e., 6a², where a is the length of the edge of a cube. The unit of the surface area of a cube and the total surface area of a cube is measured in square units, i.e., m², cm², etc. There can be two types of surface areas of a cube. They are:

• Total Surface Area of Cube
• Lateral Surface Area of Cube

Total Surface Area of Cube

The total surface area of a cube refers to the area of all the sides of the cube. Therefore, in order to find the total surface area of a cube, the sum of the area of all sides is necessary. The area of the side is the area of the square. Hence, the sum of the area of 6 square of the cube will provide the total surface area of the cube.

Lateral Surface Area of Cube

The lateral surface of a cube refers to the area of its lateral sides; the base and the top face of the cube are not included while solving for the lateral surface area of the cube. There are 4 lateral sides of the cube. The side is in the shape of a square. Therefore, four times the area of the square is the lateral surface area of the cube.

Surface Area of Cube Formula

The surface area of a cube can easily be calculated when the side length of the cube is provided. Let’s take a look at the formula for total surface area and lateral surface area of the cube,

Total Surface Area of Cube Formula

Let the length of the edge of a cube be “a” unit. Since each face of a cube is a square, the area of each face of the cube is equal to the area of a square, i.e., a². As a cube consists of 6 faces, the total surface of the cube is the Sum of the areas of the six square faces of the cube

= a² + a² + a² + a² + a² + a² = 6a²

Hence, the total surface area of a cube (TSA) = 6a²

Total surface area of a cube (TSA) = 6a² square units

Lateral Surface Area of Cube Formula

The lateral surface area of a cube is the sum of the areas of all its faces, except its top and bottom faces. Hence, the lateral surface area of the cube (LSA) is the sum of areas of all four side faces of a cube.

LSA = a² + a² + a² + a² = 4a²

Lateral surface area of the cube (LSA) = 4a² square units

Length of Edge of the Cube 

To calculate the length of edge of the cube, the surface area of the cube can be utilized. The formula for the surface area of the cube can be rearranged to find the edge of the cube.

Surface area (A) = 6a²

A = 6a²

a² = A/6

a = √A/6

Where A is the surface area of the cube and “a” is the edge length.

How to Find the Surface Area of Cube

As learned above, the lateral surface area is four times the side square, and the total surface area is six times the side square. Following are the steps that can be followed in order to find out the surface area of a cube.

Step 1: Find out the side length of the cube (Better if already given).

Step 2: Square the length/side obtained.

Step 3: In order to find the lateral surface area of the cube, multiply the squared value by 4, and in order to find the total surface area of the cube, multiply the squared value by 6.

Step 4: The value obtained is the surface area of a cube (In square units).

Solved Examples on Surface Area of Cube

Example 1: What is the total surface area of the cube if its side is 6 cm?

Solution:

Given, Side of the cube = 6 cm

The total surface area of the cube = 6a²

= 6 × 6² cm²

= 6 × 36 cm² = 216 cm²

Hence, the surface area of the cube is 216 cm².

Example 2: Find the side of a cube whose total surface area is 1350 cm².

Solution:

Given, Surface area of the cube = 1350 cm²

Let the side of the cube be “a” cm.

We know that the surface area of the cube = 6a²

⇒ 6a² = 1350

⇒ a² = 1350/6 = 225

⇒ a = √225 = 15 cm

Hence, the side of the cube = 15 cm.

Example 3: The length of the side of the cube is 10 inches. Find the lateral surface and total surface areas of a cube?

Solution:

Given, the length of the side = 10 in

We know,

The lateral surface area of a cube = 4a²

= 4 × (10)²

= 4 × 100 = 400 square inches

The total surface of a cube = 6a²

= 6 × (10)²

= 6 × 100 = 600 square inches.

Therefore, the lateral surface area of a cube is 400 square inches and its total surface area is 600 square inches.

Example 4: John is playing with a Rubik’s cube whose base area is 16 square inches. What is the length of the side of a cube, and what is its lateral surface area?

Solution: 

Given, the base area of the cube = 16 square inches

Let the length of the side of a cube be “a” inches.

We know,

The base area of a cube = a² = 16 

⇒ a = √16 = 4 inches

The lateral surface of a cube = 4a²

= 4 × 4²

= 4 × 16 = 64 square inches

Hence, the length of the side of the cube is 4 inches and its lateral surface area is 64 square inches.

Example 5: A cubical container with a side of 5 meters is to be painted on the entire outer surface area. Find the area to be painted and the total cost of painting the cube at a rate of ₨ 30 per square meter.

Solution:

Given, the length of the cubical container = 5 m

Since the area to be painted is on the outer surface, the area to be painted is equal to the total surface area of the cubical container. Hence, we need to find the total surface area of the cubical container.

Therefore, the total surface of the cubical container = 6 × (side)²

= 6 × (5)²

= 6 × 25 

= 150 square meters.

Given, the cost of painting per 1 square meter = ₨ 30 

Hence, the total cost of painting = ₨ (150 × 30) = ₨ 4500/-

Example 6: Find the ratio of the total surface area of a cube to its lateral surface area.

Solution:

Let the length of the side of a cube be “s” units.

The total surface area of the cube (TSA) = 6s²

The lateral surface area of the cube (LSA) = 4s²

Now, the ratio of the total surface area of a cube to its lateral surface area = TSA/LSA

⇒ TSA/LSA = 6s²/4s² = 3/2

Therefore, the ratio of the total surface area of a cube to its lateral surface area is 3 : 2.

FAQs on Surface Area of Cube

Question 1: What is the formula for the Surface Area of a Cube?

Answer:

Suppose the length of the side is “a” unit. Then, the formula for the total surface area of a cube is 6a^2, and the formula for the lateral surface area of the cube is 4a².

Question 2: How to find the surface area of the cube when the volume is given?

Answer:

In case the volume of the cube is given, the edge of the cube can easily be obtained through the formula. The formula for the volume of the cube is a³. Here, if the volume is given, the cube root of the volume will give “a”, that is, the edge of the cube. Now, using formula 6a², the surface area of the cube can be obtained.

Question 3: How to find the surface area of the cube with diagonals?

Answer:

The formula for the diagonals of the cube is a√3 units, where a is the side/edge of the cube. If the value of the diagonals of the cube is given, using the formula, we can first find the edge of the cube, and then, from formula 6a², we can obtain the surface area of the cube.

Question 4: How to find the formula for the base of the cube?

Answer:

The base of the cube is nothing but a square, and the formula for the area of a square is a². Therefore, the formula for the base of the cube is a².