Calculation of the Lateral Surface Area of a Cube

To find the lateral surface area (LSA) of a cube, we need to know the length of its diagonal. The LSA refers to the total area of the cube's sides, excluding the top and bottom faces.

Step 1: Understanding the Properties of a Cube

Before calculating the LSA, let's understand some essential properties of a cube:
1. All sides of a cube are equal in length.
2. The angle between any two adjacent edges of a cube is 90 degrees.
3. The diagonal of a face of the cube is the hypotenuse of a right-angled triangle formed by two adjacent edges.

Step 2: Finding the Length of a Side

To calculate the LSA, we first need to determine the length of a side of the cube. Since all sides of a cube are equal, we can use any given measurement.

Step 3: Finding the Length of Diagonal

Given the length of the diagonal, which is 6√3, we can use the Pythagorean theorem to find the length of a side. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's assume "s" as the length of a side of the cube. Then, the diagonal of a face of the cube is equal to s√2.

Using the Pythagorean theorem:
(s√2)^2 = s^2 + s^2
2s^2 = 2s^2
This equation satisfies the theorem, confirming that the diagonal of a face equals s√2.

Now, we can equate the diagonal of a face to the given diagonal:
s√2 = 6√3

Step 4: Solving for the Length of a Side

To solve for "s," we divide both sides of the equation by √2:
s = (6√3) / √2

To rationalize the denominator, we multiply both the numerator and denominator by √2:
s = (6√3 * √2) / 2
s = (6√6) / 2
s = 3√6

Step 5: Calculating the Lateral Surface Area

The LSA of a cube is given by the formula:
LSA = 4s^2

Substituting the value of "s" we found earlier:
LSA = 4(3√6)^2
LSA = 4(9 * 6)
LSA = 4 * 54
LSA = 216 square units

Final Answer:
The lateral surface area of the cube with a diagonal length of 6√3 is 216 square units.