Question

The diagonal of a cube is \(9\sqrt{3}\) cm. Find the total surface area of cube.

Hint:

Remember the difference between a face diagonal and a space diagonal of a cube. A face diagonal has length \(a\sqrt{2}\), while a space diagonal (connecting opposite vertices) has length \(a\sqrt{3}\). Read the question carefully to know which one is given.

Answer 1

Step 1: Understanding the Concept:
This problem connects three properties of a cube: its side length (a), its space diagonal (d), and its total surface area (TSA). We need to use the formula for the diagonal to find the side length first, and then use the side length to find the TSA.

Step 2: Key Formula or Approach:
1. The length of the space diagonal of a cube with side length \(a\) is given by \(d = a\sqrt{3}\).
2. The Total Surface Area (TSA) of a cube is the sum of the areas of its six square faces, given by \(TSA = 6a^2\).

Step 3: Detailed Explanation:
Given: Diagonal of the cube, \(d = 9\sqrt{3}\) cm.
First, find the side length \(a\) of the cube using the diagonal formula: \[ a\sqrt{3} = 9\sqrt{3} \] Divide both sides by \(\sqrt{3}\): \[ a = 9 \text{ cm} \] Now that we have the side length, we can calculate the Total Surface Area: \[ TSA = 6a^2 \] \[ TSA = 6 \times (9)^2 \] \[ TSA = 6 \times 81 \] \[ TSA = 486 \text{ cm}^2 \]

Step 4: Final Answer:
The total surface area of the cube is 486 cm\(^2\).