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Surface Area of Cube

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Surface area of cube is defined as the total area covered by all the faces 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. Since the six faces of a cube are squares, the length, breadth, and height of a cube are equal. Thus, the surface area of the cube is six times the area of a square. Let’s learn more about the Surface Area of the Cube, its formula, and others in detail in this article.

Surface Area of Cube Definition

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
Surface Area of Cube

 

Total Surface Area of Cube

The total surface area of a cube refers to the area of all the faces of the cube. Therefore, in order to find the total surface area of a cube, the sum of the area of all faces is necessary. The area of the faces is the area of a square as each face of the cube is square. Hence, the sum of the area of 6 squares 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 faces of the cube, and as we know, each face is 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 the 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.

TSA = 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 

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 the areas of all four side faces of a cube.

LSA = a2 + a2 + a2 + a2 = 4a2

Lateral surface area of the cube (LSA) = 4a

Length of Edge of the Cube 

To calculate the length of the 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

Length of Edge of Cube  = √A/6

Where A is the total surface area of the cube.

How to Find the Surface Area of a 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).

Surface Area of Cube (when Volume is given)

The surface area of the cube is calculated using the formula, 

Surface Area of Cube = 6a2

And we know, the formula for the volume of a cube.

Volume of Cube = Side3

⇒ Side of Cube (a) = 3√(Volume of Cube)

Using this formula we get the side of the cube and then the surface area is calculated using the side, or we can use the direct formula given below:

Surface Area = 6 × (Volume of Cube)2/3

Example: Find the surface area of a cube whose volume is 643 cubic units.

Solution:

Volume of cube (a)3 = 643

a = 3√(643)

⇒ a = 7 units.

Thus, Surface Area of Cube = 6a2 

⇒ Surface Area of Cube = 6(7)2 

⇒ Surface Area of Cube = 294 sq. units

Surface Area of Cube (when Diagonal is given)

Surface area of the cube is calculated using the formula,

Surface Area = 6a2

If the diagonal of the cube is given then its side is calculated using the formula.

Diagonal = √3a

Side of Cube (a) = Diagonal/√(3)

Using this formula we get the side of the cube and then the surface area is calculated using the side or we can use the following formula:

Surface Area = 2(Diagonal)2

Example: Find the surface area of the cube when the diagonal is 8√3 units.

Solution:

Diagonal of Cube (√3a) = 8√3

Solving the above equation,

a = 8√3/√3 = 8 unit

Surface Area of Cube = 6a2 

⇒ Surface Area of Cube = 6(8)2 

⇒ Surface Area of Cube = 288 square units.

Net Of Cube

The net of any 3-D figure is the 2-D representation of that 3-D figure. For a cube, we have six equal faces in its nets and each of the following faces represents a square.

We know that a cube has six faces, and each face is a square. Thus, the area of one face with side “a”

Area = a2

Total Surface Area of Cube = 6a2

The net of the cube is given in the image below,

Net Of Cube

 

Surface Area of Cube and Cuboid

Cube is a 3-dimensional figure made of Six square faces then the formula for the surface area of a cube,

  • TSA of Cube = 6a2
  • CSA of Cube = 4a2

where a is the side of the cube.

Cubiod is a 3-dimensional figure made of six rectangles of different dimensions than the formula for surface area of a cuboid,

  • TSA of Cube = 2(lb + bh + lh)
  • CSA of Cube = 2h(l + b)

where l, b and h are the length, breadth and height of the cuboid respectively.

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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

Total Srface Area of Cube = 6a2

                                         = 6 × 62 cm2

                                         = 6 × 36 cm

                                                       = 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 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,

Lateral Surface Area of a cube = 4a2

                                                = 4 × (10)2

                                                = 4 × 100 = 400 square inches

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: Base Area of Cube = 16 square inches

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

We know,

Base area of a cube = a2 = 16 

a = √16 = 4 inches

Lateral Surface of a Cube = 4a2

⇒ Lateral Surface of a Cube = 4 × 42

⇒ Lateral Surface of a Cube = 4 × 16 

⇒ Lateral Surface of a Cube = 64 sq. 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 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.

Total Surface Area of cubical container = 6 × (side)2

⇒ TSA = 6 × (5)2

⇒ TSA  = 6 × 25 

⇒ TSA  = 150 sq meters.

Given,

Cost of Painting = ₨ 30 per square meter 

Hence, 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.

Total surface area of the cube (TSA) = 6s2

Lateral surface area of the cube (LSA) = 4s2

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

⇒ Required Ratio = 6s2/4s2 

⇒ Required Ratio = 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

Q1: What is Surface Area of Cube?

Answer:

The surface area of the cube is the total area required to cover the cube completely. As each face of the cube is square and it has a total of six faces then its surface area is six times the area of one face.

Q2: What is the formula for Surface Area of a Cube?

Answer:

Suppose the side length of the cube be ‘a’ then its surface area is calculated using the formula,

  • Total Surface Area of Cube = 6a2
  • Lateral Surface Area of Cube = 4a2

Q3: What is lateral surface area of cube?

Answer:

The lateral surface area of the cube is the area required to cover the cube laterally leaving its base and top faces. Lateral surface area of the cube is also called Curved Surface Area(CSA)

CSA of Cube = 4a2

where a is the side of the cube.

Q4: What is the total surface area of a cube?

Answer:

The total surface area of the cube is the area required to cover the cube completely including its base and top faces. Total surface area of the cube is calculated using the formula

TSA of Cube = 6a2

where a is the side of the cube.

Q5: What is the surface area of cube and cuboid?

Answer:

The formula for surface area of cube,

  • TSA of Cube = 6a2
  • CSA of Cube = 4a2

where a is the side of the cube.

The formula for surface area of cuboid,

  • TSA of Cube = 2(lb + bh + lh)
  • CSA of Cube = 2h(l + b)

where l, b and h are the length, breadth and height of the cubiod respectively.

Q6: How to find surface area of cube with volume?

Answer:

Formula for Volume of cube = a3, where a is the side of the cube.

If volume(V) is given then the side is calculated as,

Side of cube (a) = 3√(V)

Then the surface area is calculated using the formula,

TSA = 6a2

Q7: How to find surface area of cube with diagonals?

Answer:

Formula for Diagonal of cube = √3a, where a is the side of the cube.

If diagonal(d) is given then the side is calculated as,

Side of cube (a) = d/√(3)

Then the surface area is calculated using the formula,

TSA = 6a2



Last Updated : 24 Jan, 2024
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