Understanding the Cube and its Surface Area

A cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. All 12 edges are of the same length. The total surface area of a cube is the sum of the areas of all six of its square faces.

Calculating Total Surface Area of a Cube

Let the side length of the cube be denoted by 'a'. Since each face is a square with side 'a', the area of one face is given by the formula for the area of a square, which is \(\text{side} \times \text{side} = a \times a = a^2\).

A cube has 6 identical square faces. Therefore, the total surface area (TSA) of a cube is the sum of the areas of these 6 faces.

The formula for the total surface area (TSA) of a cube is:

\(\text{TSA} = 6 \times (\text{Area of one face})\)

\(\text{TSA} = 6 \times a^2\)

Finding the Side of a Cube from Total Surface Area

We are given that the total surface area of the cube is 96 sq. cm. We can use the formula for the total surface area to find the side length 'a'.

Given TSA = 96 sq. cm.

Using the formula:

\(6a^2 = \text{TSA}\)

\(6a^2 = 96\)

To find \(a^2\), we need to divide the total surface area by 6:

\(a^2 = \frac{96}{6}\)

Now, let's perform the division:

\(96 \div 6 = 16\)

So, \(a^2 = 16\)

To find the side length 'a', we need to take the square root of \(a^2\). Since 'a' represents a length, it must be a positive value.

\(a = \sqrt{16}\)

\(a = 4\)

The side length of the cube is 4 cm.

Verifying the Side Length of the Cube

Let's check if a cube with a side length of 4 cm has a total surface area of 96 sq. cm.

Side length (a) = 4 cm

Area of one face = \(a^2 = (4 \text{ cm})^2 = 16 \text{ sq. cm}\)

Total Surface Area (TSA) = \(6 \times (\text{Area of one face})\)

TSA = \(6 \times 16 \text{ sq. cm} = 96 \text{ sq. cm}\)

This matches the given total surface area in the question.

Comparing with Given Options

We found the side length of the cube to be 4 cm. Let's look at the given options:

• 3 cm
• 5 cm
• 4 cm
• 2 cm

Our calculated side length, 4 cm, matches one of the options.

Conclusion on Cube Side Length

Based on the calculations, if the total surface area of a cube is 96 sq. cm, its side length is 4 cm.

Revision Table: Cube Formulas

Additional Information on Cube Properties

Cubes are fundamental shapes in geometry and have several interesting properties:

• A cube is a special type of rectangular prism (cuboid) where all edges are equal.
• It is one of the five Platonic solids, known for having identical regular polygonal faces, the same number of faces meeting at each vertex, and the same angle between any two adjacent faces.
• The angle between any two adjacent faces of a cube is 90 degrees.
• The total number of edges in a cube is 12.
• The total number of vertices in a cube is 8.
• The total number of faces in a cube is 6.

Understanding the relationship between the side length and the total surface area is crucial for solving problems involving cubes, especially in geometry and measurement topics.