Surface Area of a Cube Formula

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Surface Area Of Cube

Surface area of cube is the sum of areas of all the faces of cube, that covers it. The formula for surface area is equal to six times of square of length of the sides of cube. It is represented by 6a2, where a is the side length of cube.It is basically the total surface area. Also, learn Volume Of A Cube.

When a cube is kept in a three-dimensional space, the area occupied by the sides of the cube in the space is called surface area of cube. In real world, we have been surrounded by many solid objects which have their own area as well volume. The area defines the region occupied by the objects and volume defines the space contained inside that object. The basic solid shapes or 3D shapes which we have learned till now in geometry are:

  • Cube
  • Cuboid
  • Cylinder
  • Cone
  • Sphere

Surface Area of Cube Definition

The definition of surface area of a given cube states that if the total surface area is equal to the sum of all the areas of the faces of the cube. Since the cube has six faces, therefore, the total surface area of the cube will be equal to sum of all six faces of cube.

Since, the surface of the cube is in square shape. Hence, area of each face of the cube is equal to square of edge. Let the length of edge of cube is a.

Therefore, area of one face = a2    [By area of square formula]

There are total 6 faces. Therefore,

TSA of cube = a2 + a2 + a2 + a2 + a2 +a2

TSA of cube = 6a2

A cube consists of ‘n’ number of square units. Hence the space covered by these square units on the surface of the cube is the surface area. Basically, the surface area is the sum of all the area of all the shapes that cover the surface of the shape or object. In the case of a cube, there are 6 faces. So the surface area will be sum of all the area of six faces.

Let us derive the formula for surface area for a given cube, to solve problems based on it.

Surface Area of a Cube Formula

As per the definition of the cube, we know, the cube consists of 6 square faces. Let us consider, a cube whose length of the edges is ‘a’.

Now, we know, by the formula of area of a square;

Area = Side= a2

Therefore, the total surface area of a cube = 6 × (area of each side)

= 6 × a2 = 6a2 Square Unit

TSA = 6 a2

Length of Edge of the Cube

From the formula of the surface area of the cube, we can also find the length of the edge of the cube by rearranging the formula, such as;

A = 6 (side)2

side2 = A/6

side = √(A/6) = Edge length

where A is the area.

Examples

Q.1: Calculate the cost required to paint an aquarium which is in cube shape having an edge length of 10m. If the painting cost of an aquarium is INR 3/m2.

Solution: Total surface area of aquarium = 6 (side)2

= 6 (10)2

= 600 sq.m

Total cost of painting the aquarium = 3 × 600 = Rs. 1800

Q.2: If the sidewall of a cubic structure have length 7m, then find the total surface area.

Solution: Given, the length of the sidewall = 7m

As per the formula, we know;

TSA = 6a2

TSA = 6 x 7 x 7 = 294 sq.m

Q.3: Find the length of the edge of the cube, if its area is 2400 sq.cm.

Solution: Given, area = 2400 sq.cm.

We know,

Length of edge of cube = √(A/6) = √(2400/6) = √400 = 20 cm.

To know more about Geometry and surface area of different shapes, you can visit BYJU’S.

Frequently Asked Questions – FAQs

How to find the surface area of cube?

Each face of the cube is in square shape. We know the area of square is equal to side2. Since, there are six faces of cube, therefore the total surface area or surface area of a cube will be equal to sum of areas of all its surfaces, i.e.6side2.

What is the formula of surface area of cube?

If a is the edge length of a surface of a cube, then the formula of surface area is given by 6a2.

What is the volume of a cube?

The volume of a cube is the space contained by a cube. It is calculated by the formula a3, where a is the edge length of the cube.

What is the surface area of cube?

The surface area is the region occupied by the surfaces of the cube in a three-dimensional space. The six faces of the cube occupy some space which we need to find using the formula, i.e.6a2, where a is edge length of the cube.

What is the lateral surface area of cube?

The lateral surface area of cube is the total surface area of the cube, which is equal to sum of areas of all its sides.

What is the Surface Area of Cube?

The surface area of the cube will be the sum of the area of the bases and the area of lateral surfaces 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 the surface area of one face added six times to itself. It is measured as the “number of square units” (square centimeters, square inches, square feet, etc.). The surface area of a cube can be of two types,

  • Lateral surface area
  • Total surface area

Total Surface Area of Cube

The total surface area of a cube refers to the total area covered by all six faces of a cube. To calculate the TSA of a cube, we find the sum of the areas of these 6 faces.

Lateral Surface Area of Cube

The lateral surface area of a cube refers to the total area covered by the side or lateral faces of a cube. To calculate LSA, we find the sum of areas of these 4 faces.

Surface Area of Cube Formula

The surface area of a cube can be calculated given the edge length. Let us understand the formula for the lateral and total surface area of a cube.

Total Surface Area of Cube Formula

The formula of the total surface area of the cube is used to find the area occupied by the six surfaces. TSA of the cube is obtained by multiplying the square of its side length by 6. Thus, the formula for the surface area of the cube, with side length “a” is “6a2“.

Total Surface Area of a Cube = (6 × side2) square units

Lateral Surface Area of Cube Formula

The formula of the lateral surface area of the cube is used to find the area occupied by the four lateral or side surfaces. LSA of the cube is obtained by multiplying the square of its side length by 4. Thus, the formula for the lateral surface area of the cube, with side length “a” is “4a2“.

Lateral Surface Area of a Cube = (4 × side2) square units

How to Find Surface Area of Cube?

The total surface area of a cube is equal to the square of its side length times 6. Similarly, for lateral surface area, we multiply the square of side length by 4. By following the steps mentioned below, we can find the surface area of the cube:

  • Step 1: Identify the length of the side of the cube.
  • Step 2: Find the square of the length of the side of the cube.
  • Step 3: For total surface area, find out the product of the square of side length by 6, while for lateral surface area multiply the product of the square of side length by 4.
  • Step 4: Write your answer in square units.

Surface Area of Cube Examples

Example 2: Olive has been given a cube of base area 64 square units. Find the length of the side of the cube and the total surface area of the cube.

Solution:

The base area of the cube = 64 square units.

Length of the side of the cube ‘a’ = √64 = 8 units.

Total surface area: A = 6a2

A = 6 × 82

A = 384

Therefore, the length of the base of the cube is 8 units, and the area of the cube is 384 square units.

Example 3: What is the lateral surface area of a cube of side length 12 feet?

Solution:

Given, side length (a) = 12 feet.

Lateral surface area: L = 4a2

L = 4 × 122

L = 576

Therefore, the lateral surface area of the cube is 576 square feet.

FAQs on Surface Area of Cube

What is the Surface Area of a Cube?

The surface area of a cube is defined as the total area covered by the faces of a cube. To calculate the surface area of a cube, we find the sum of the area of all the faces of a cube.

What is the Formula of Surface Area of Cube?

The surface area of a cube with edge length as ‘a’ can be calculated using the following formulas, LSA of Cube = 4a2 square units and, TSA of Cube = 6a2 square units.

What is the Unit Used to Express Surface Area of Cube?

The surface area of a cube is expressed in square units, for example using units like in2, ft2, yd2, m2, cm2, etc.

What is the Lateral Surface Area of a Cube?

The lateral area of a cube is the total area covered by the lateral or side faces of a cube. The formula to calculate the lateral surface area of a cube is given as, Lateral surface area = 4a2, where, ‘a’ is the edge length of the cube.

How to Find Total Surface Area of Cube?

The total surface area of a cube is the area covered by all six faces of a cube. The formula to find the total surface area of a cube is given as, Total surface area = 6a2, where, ‘a’ is the edge length of the cube.

How to Find Surface Area of Cube When Volume is Given?

When volume is given, we first find the length of one side of the cube and then apply the formula for the surface area of the cube. The volume of the cube formula is (side)3 which can be used to find the side length. For example, if the volume of a cube is 64 cubic units, then the length of one side of the cube = 3√64 = 4 units. Now, by using the surface area of the cube formula, i.e 6 × (side)2, we can find its surface area. This implies, TSA = 6 × 4 × 4 = 96 square units.

How to Find Surface Area of Cube with Diagonal?

The formula of diagonal of a cube is a√3 units, where a is the length of one side of the cube. By using this formula and the given value of diagonal, we can first find the side length of the cube followed by finding its surface area.

What is the Formula to Find the Area of the Base of a Cube?

The base of the cube is in the shape of a square. The formula to find the area of the base of a cube is a2, where a is the length of the side of the cube.

Definition

Surface Area of a cube is the total area of the outside surfaces of the cube and is given by A= 6a2, where a is the edge.

A cube has 6 identical square faces and hence it is also called as a hexahedron. Each face of a cube has 4 edges and totally there are 12 edges. It is measured in terms of square unit.

How to Find the Surface Area of a Cube

The Surface Area of a Cube Formula is,

Surface area of Cube = 6a2where “a” is the side length of the cube.

Solved Examples

Question 1: What is the surface area of a cube of side 5 cm?

Solution:

Given,
Side of the cube = a = 5 cm
Surface Area of a Cube = 6a2
= 6 × 52 cm2
= 6 × 25 cm2
= 150 cm2
Question 2: What is the surface area of a cube of side 17 cm?

Solution:

Given,
Side of the cube = a = 17 cm
Surface Area of a Cube = 6a2
= 6 × 172 cm2
= 6 × 289 cm2
= 1734 cm2

Question 3: Find the side of a cube whose surface area is 1014 cm2.

Solution:

Given surface area of a cube = 1014 cm2

Let “a” be the side of the cube.
We know that,

Surface area of cube = 6a2

⇒ 6a2 = 1014

⇒ a2 = 1014/6

⇒ a2 = 169

⇒ a = √169

⇒ a = 13

Therefore, side of the cube = 13 cm

  1. What is the surface area of a cube of side equal to 5 cm?

Solution: Given,

Side of the cube equals a = 5 cm

Total Surface Area of Cube formula = 6a2

= 6 × 5 × 5 

= 6 × 25 cm2

= 150 cm2

  1. What is the surface area of a cube of side equal to 17 cm?

Solution: Given,

Side of the cube equals a = 17 cm

Total Surface Area of Cube formula = 6a2

= 6 × 172cm2

= 6 × 289 cm2

= 1734 cm2

  1. Calculate the cost of painting an aquarium that is in the shape of a cube with a 10m edge length. If the cost of painting an aquarium is INR 3/m*2, how much does it cost to paint an aquarium?

Solution: The aquarium’s total surface area is 6 square feet (side)^ 2

=6102

= 600 square meters

The total cost of painting the aquarium  = 3 × 600 = Rs. 1800

  1. Find the overall surface area of a cubic construction if the sidewalls are 7 meters long.

Solution: Assume the sidewall is 7 meters long.

We know what the formula says;

TSA = 6a2

6 x 7 x 7 = 294 sq.m = TSA 

FAQs on Surface Area of a Cube Formula

1. How do you calculate a cube’s surface area?

To find the surface area of a cube, multiply the size of one of the square sides by 6, because there are six sides. This is equivalent to solving with the formula Total surface Area of Cube = 6s2

2. What is the Surface Area Formula?

Surface area can be defined as the sum of the areas of all faces (or surfaces) on a 3D shape. A cuboid has 6 rectangular faces. To find the surface area of a cuboid, add the areas of all 6 faces. Taking the length as (l), width as (w), and height as (h) of the prism and the formula, SA = 2lw + 2lh + 2hw, to find the surface area.

3. What is the distinction between a cube and a cuboid?

The main distinction between a cube and a cuboid is that a cube has six identical square-shaped faces, whereas a cuboid has rectangular faces. Although the structure of a cube and a cuboid are similar, they have a few differences in terms of edge length, diagonals, and faces.

  1. What is a Cube?

A cube is a three-dimensional object with six square faces that are all the same size. All six square faces of the cube have the same dimensions. A regular hexahedron or a square prism are two other names for a cube. A cube can be anything from an ice cube to a Rubik’s cube to a conventional dice. There are six square faces, eight vertices, and twelve edges in this puzzle. make up a cube. 

What is a Cube?

The only regular hexahedron, a cube is a three-dimensional object with six equal-sized square surfaces or sides, 12 edges, and 8 vertices. Given that its square sides are equal, it follows that a cube’s length, width, and height are equal, too. Examples of cube-shaped objects are dice, jewelry boxes, ice cubes, sugar cubes, and Rubik’s cubes.

Here’s an illustration of a cube. Notice how it forms 6 equal square surfaces or sides when unfolded. The resulting two-dimensional shape when a cube is unfolded is called the cube’s net.

How to Find the Surface Area of a Cube:

Recall that the surface area of a three-dimensional figure refers to the total area occupied by the figure’s surface. To better understand surface area, look at the net or the flat layout of the cube in the illustration below. 

The surface area of a cube is the sum of all the areas of its 6 square sides. Recall that the area of a square is computed by multiplying the length of each side (a) by itself: a • a  or a². Just multiply the area of a square side by 6 and you’ll have the cube’s surface area.

Use this formula to find the total surface area of a cube: SA = 6a²
where a = length of the cube’s one side and a² = area of one square side of the cube.

Note: Don’t forget that surface areas are measured in square units such as cm2, m2, km2, and in2.

Quick Guide to Finding the Surface Area of a Cube:

Step 1. Write the given figures. You’ll need the length of the cube’s side to find the surface area. Make sure all measurement units are the same. If not, convert either of them to match the other. 

Step 2. Plug the figures into the formula.

Step 3. Perform the operations. Don’t forget to write the square unit for surface area.

Example #1:

Find the surface area of the cube below.

Solution for Example #1:

Step 1. Write the given measurement, a = 8m.
Step 2. Substitute 8m for a in the formula for the surface area.
SA = 6a²
SA = 6(8m)²
Step 3. Simplify & solve the equation.
SA = 6(64m²)
SA = 384m²

Therefore, the surface area is 384m².

Example #2:

Find the surface area of a cube with a length of 4m.

Step 1. Write the given measurement, a = 4m.
Step 2. Substitute 4m for a in the formula for the surface area.
SA = 6a²
SA = 6(4m)²
Step 3. Simplify & solve the equation.
SA = 6(16m²)
SA = 96m²

Therefore, the surface area is 96m².

Thank you for reading. We hope it’s effective! Always feel free to revisit this page if you ever have any questions about the surface area of a cube.

Example:

Determine the surface area of the cube.

Solution:

If working with fraction is difficult for you, convert to a decimal!

The surface area of this cube is 37.5 square inches.

If you know the surface area you can also work backwards to determine the side lengths.

Example: The surface area of a cube is 86.64 square meters. Determine the side length of the cube.

Solution: Because we already know the surface area, we can work backwards through the steps to determine the length of the side.

The side length was squared and then multiplied by 6. So we will divide by 6 to undo the multiplying. Then we will take the square root to undo the squaring.

Therefore, the sides of the cube are each 3.8 meters long.
Example: The surface area of a cube is 384 m2. Determine the volume of the cube. Solution: We need to know the side length in order to determine the volume. So the first step is to work backwards from the surface area of the side length.

The side length of the cube is 8 meters.
Now use this value to determine the volume using the formula V = s3.

The volume of the cube is 512 cubic meters.

The surface area of the block is 6 x 9, or 54 square centimeters.

The same formula can be used to find the length of one side of a cube if given the surface area of the cube.

Frequently Asked Questions

What is the total surface area of a cube?

The total surface area of a cube is the area of the space covering the outside surface of the cube. The surface area of a cube can be found by finding the area of one of the six congruent square faces of the cube, and then multiplying that area by 6. The formula is SA = 6s^2.

How do you find the area of a cube?

The area of a cube is the sum of the areas of the six congruent square faces that cover the surface of the cube. The area of a cube can be found using the formula SA = 6s^2, where SA is the surface area of the cube, in square units, and s is the length of one side of the cube, in units.

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