Cube Difference Calculation

This problem requires finding the difference between the total surface area and the lateral surface area of a cube with a given side length of 11 cm.

Understanding the terms:

• A cube is a 3D shape with six identical square faces.
• Total Surface Area (TSA) refers to the sum of the areas of all six faces.
• Lateral Surface Area (LSA) refers to the sum of the areas of the four vertical faces, excluding the top and bottom faces.

Cube Surface Area Formulas

Let '$a$' represent the length of one side of the cube. The formulas for the surface areas are:

• Total Surface Area (TSA): $TSA = 6a^2$
• Lateral Surface Area (LSA): $LSA = 4a^2$

Deriving the Difference Formula

The question asks for the difference between TSA and LSA.

Difference = TSA - LSA

Substituting the formulas into the equation:

Difference = $6a^2 - 4a^2$

By simplifying the expression, we get:

Difference = $2a^2$

This result indicates that the difference in surface area is equivalent to the area of two faces of the cube.

Applying Side Length 11 cm

We are given that the side length of the cube is $a = 11$ cm.

Using the simplified formula for the difference, $2a^2$, we can calculate the value:

Difference = $2 \times (11 \text{ cm})^2$

First, calculate the square of the side length:

$(11 \text{ cm})^2 = 11 \times 11 \text{ cm}^2 = 121 \text{ cm}^2$

Next, multiply this result by 2:

Difference = $2 \times 121 \text{ cm}^2 = 242 \text{ cm}^2$

Calculation Steps & Verification

To confirm the result, we can calculate the TSA and LSA separately using the given side length ($a = 11$ cm).

1. Calculate Total Surface Area (TSA):

$TSA = 6a^2 = 6 \times (11 \text{ cm})^2 = 6 \times 121 \text{ cm}^2 = 726 \text{ cm}^2$

2. Calculate Lateral Surface Area (LSA):

$LSA = 4a^2 = 4 \times (11 \text{ cm})^2 = 4 \times 121 \text{ cm}^2 = 484 \text{ cm}^2$

3. Compute the Difference:

Difference = $TSA - LSA = 726 \text{ cm}^2 - 484 \text{ cm}^2 = 242 \text{ cm}^2$

Both calculation methods confirm that the difference is 242 square centimeters.