Surface Area of Cube

The surface area of the cube, a six-faced three-dimensional object, is defined as 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 faces. A cube is a three-dimensional solid figure which consists of square faces.

The surface area of a cube is an important geometric measurement and is used in many real-life applications, such as architecture, engineering, and manufacturing. For example, architects use the surface area of a cube to determine the amount of material required to construct a building or a room, while manufacturers use it to calculate the amount of paint or other coatings needed to cover the surface of a cube-shaped object. Let us learn the surface area of cube formula along with how it is derived.

The surface area of the cube is be the sum of the area of the bases and the areas of the lateral surfaces of the cube. Since all six faces of the cube are made up of squares of the same dimensions, 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 centimetres, square inches, square feet, etc.). The surface area of a cube can be of two types,

• Lateral Surface Area (LSA) (or) Curved Surface Area (CSA) - The sum of areas of side faces
• Total Surface Area (TSA) - The sum of areas of side faces + The sum of areas of two bases

TSA of a Cube

The TSA (total surface area) of a cube refers to the total area covered by all six faces of a cube. To calculate the TSA, we find the sum of the areas of these 6 faces. Note that the TSA of a cube is often referred to as just surface area (or) area of cube

LSA of a Cube

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

The surface area of a cube can be calculated given the edge length (a). Here are the surface area of cube formulas:

• The TSA of cube formula = 6a²
• The CSA (or) LSA of cube formula = 4a²

Let us understand the formula for the lateral and total surface area of a cube.

Total Surface Area of Cube (TSA) 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 (LSA) 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

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 the 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 and denote it as 'a'.
• Step 2: Find the square of the length of the side of the cube. i.e., a².
• Step 3: For total surface area, multiply a² by 6, while for lateral surface area multiply a² by 4.
• Step 4: Write your answer in square units.

But sometimes, the edge length of the cube might not have been given. Instead, the volume (or) diagonal might be given. Let us see how to find the surface area of cube in such cases:

Surface Area of Cube Given Volume

The volume of a cube of side length 'a' is given by a³. Thus, when the volume of cube is given, take its cube root to find 'a'. Then, substitute the value of 'a' in the formula 6a² to find the surface area of the cube.

Example: What is the surface area of cube whose volume is 125 cubic units?

Solution:

The volume is, a³ = 125. This gives, a = ³√125 = 5 units.

Then surface area = 6a² = 6(5)² = 150 square units.

Surface Area of Cube Given Diagonal

The diagonal of a cube of side length 'a' is given by the formula a√3. When we know the diagonal of a cube, set it equal to a√3 and solve it for 'a'. Then, we can find the surface area using the formula 6a².

Example: Find the surface area of a cube whose diagonal is 10√3 units.

Solution:

The diagonal of the cube is a√3 = 10√3. Dividing both sides by √3, we get a = 10 units.

Surface area of cube = 6a² = 6(10)² = 600 square units.

Important Points on Surface Area of Cube:

• The surface area of a cube is nothing but the TSA of the cube.
• Thus, surface area of cube formula = TSA = 6a², where 'a' is the side length of the cube.
• CSA (or) LSA of cube = 4a²

☛Related Articles:

Check these interesting articles related to the concept of the surface area of the cube.

• Surface Area of a Cube Calculator
• Surface Area of Cuboid
• Difference Between Area and Surface Area
• Surface Area Formulas

What Does the Surface Area of a Cube Mean?

The surface area of a cube means 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. If 'x' is the side length of the cube then its area of cube = 6x².

What is the Area of Cube Formulas?

The surface area of a cube with edge length as 'a' can be calculated using the following formulas:

1. LSA of Cube = 4a² square units
2. TSA of Cube = 6a² square units.

Note that LSA of cube is sometimes referred to as CSA; and TSA means just the surface area of cube.

What is the Curved Surface Area of Cube Formula?

The curved area of a cube is the total area covered by the lateral or side faces of a cube. It is commonly abbreviated as CSA. The formula to calculate the curved surface area of a cube is given as, CSA = 4a², where, 'a' is the edge length of the cube.

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

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

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.

How to Find Total Surface Area of Cube?

The total surface area (TSA) 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, TSA = 6a², where, 'a' is the side length of the cube.

How to Find the 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 surface area formula 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.

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 (which is nothing but area of square formula) is a², where a is the length of the side of the cube.