Surface Area of a Cube Formula and Solved Examples

The Surface Area of a cube formula, a three-dimensional object with six faces, is defined as the collective area encompassed by all six sides of the cube. Calculating the total surface area of a cube involves determining the area of its two base faces as well as the area of the four lateral faces. A cube, characterized by its six congruent square faces, is a solid figure in three-dimensional space.

The measurement of the surface area of a cube holds significance within the realm of geometry and finds practical application in various real-life scenarios, including architecture, engineering, and manufacturing. For instance, architects leverage the cube's surface area to ascertain the quantity of materials essential for the construction of buildings or rooms. Similarly, manufacturers employ this measurement to compute the quantity of paint or coatings required to cover the surface of objects shaped like cubes. In this regard, understanding the formula for calculating the surface area of a cube is pivotal, and we shall delve into its derivation and application.

What is the Surface Area of a Cube Formula?

The surface area of a cube is calculated by combining the area of its bases with the area of its lateral surfaces. Given that all six faces of the cube are congruent squares of identical dimensions, determining the total surface area entails adding the area of one face to itself six times. This measurement is quantified in terms of "square units" (like square centimetres, square inches, or square feet). The surface area of a cube encompasses two variations:

• Lateral Surface Area (LSA) or Curved Surface Area (CSA): This is the summation of the areas of the cube's side faces.
• Total Surface Area (TSA): It includes the combined areas of the side faces as well as the areas of the two base faces.

In essence, the surface area of a cube constitutes a vital geometric measure that underlies various calculations and practical applications.

Total Surface Area of a Cube

The total surface area (TSA) of a cube pertains to the cumulative extent occupied by all six faces of the cube. To compute the TSA, the summation of the areas of these six faces is determined. It's worth noting that the TSA of a cube is frequently denoted as the surface area or the area of the cube itself.

Lateral Surface Area of a Cube

The lateral surface area (LSA) of a cube designates the collective region encompassed by the four lateral faces of the cube. To ascertain the LSA, the aggregate area of these four faces is calculated

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Cube Surface Area Formula

Calculating the surface area of a cube involves using the edge length (a) as a parameter. Here are the formulas for cube surface area:

• The formula for the Total Surface Area (TSA) of a cube: TSA = 6a²
• The formula for the Curved Surface Area (CSA) or Lateral Surface Area (LSA) of a cube: CSA = 4a²

Let's explore the derivation and significance of these formulas for both the lateral and total surface area of a cube.

The Formula for the Total Surface Area of the Cube (TSA)

The formula to determine the total surface area of a cube serves to quantify the space covered by all six faces. The TSA of the cube can be obtained by multiplying the square of its side length by 6. Mathematically, the formula for the surface area of a cube with a side length "a" is expressed as "6a²".

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

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Formula for Lateral Surface Area of Cube (LSA)

The formula for calculating the lateral surface area of a cube is aimed at evaluating the region occupied by its four lateral or side surfaces. To compute the LSA of the cube, the square of its side length is multiplied by 4. Consequently, the formula for the lateral surface area of a cube with a side length "a" translates to "4a²".

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

Determining the Surface Area of a Cube

To ascertain the total surface area of a cube, the square of its side length is multiplied by 6. Similarly, for the lateral surface area, the square of the side length is multiplied by 4. To calculate the surface area of the cube, follow these steps:

Step 1: Identify the length of a side of the cube, denoting it as 'a'.

Step 2: Square the length of the cube's side (a²).

Step 3: For the total surface area, multiply a² by 6; for the lateral surface area, multiply a² by 4.

Step 4: Present your answer in terms of square units.

However, in certain instances, the edge length of the cube might not be directly provided. Instead, you might have information about the cube's volume or diagonal. Let's explore how to determine the cube's surface area under such circumstances:

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Surface Area of Cube Given Volume

When the volume of a cube with a side length 'a' is known (volume = a³), calculate the cube root of the volume to find 'a'. Subsequently, substitute the value of 'a' into the formula 6a² to deduce the surface area of the cube.

Example: Determine the surface area of a cube with a volume of 125 cubic units.

Solution:

Given volume: a³ = 125. This implies, a = ³√125 = 5 units.

Hence, the surface area = 6a² = 6(5)² = 150 square units.

Surface Area of Cube Given Diagonal

If the diagonal of a cube with a side length 'a' is represented by the formula a√3, solve for 'a' by equating it to the given diagonal. Subsequently, utilize the value of 'a' in the formula 6a² to compute the cube's surface area.

Example: Calculate the surface area of a cube with a diagonal of 10√3 units.

Solution:

Given diagonal: a√3 = 10√3. Dividing both sides by √3, we deduce a = 10 units.

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

Key Points on Cube Surface Area

• The surface area of a cube is synonymous with its TSA.
• Thus, the formula for cube surface area equates to TSA = 6a², where 'a' represents the side length of the cube.
• The CSA or LSA of a cube amounts to 4a².