Question

Diagonal of a Cube is \(\sqrt{6}\) cm., then its lateral surface area is :

• A. \(6\sqrt{6} \text{ cm}^2\)
• B. \(36 \text{ cm}^2\)
• C. \(12 \text{ cm}^2\)
• D. \(8 \text{ cm}^2\)

Hint:

1. Remember the formula for the diagonal of a cube: \(d = a\sqrt{3}\), where \(a\) is the side length. 2. Use the given diagonal (\(\sqrt{6}\)) to find \(a\): \(\sqrt{6} = a\sqrt{3} \Rightarrow a = \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{2}\) cm. 3. Remember the formula for the lateral surface area of a cube: LSA = \(4a^2\) (area of 4 side faces). 4. Calculate LSA: \(4 \times (\sqrt{2})^2 = 4 \times 2 = 8 \text{ cm}^2\).

Answer 1

The Correct Option is D

Concept: (A) The diagonal (\(d\)) of a cube with side length \(a\) is given by the formula \(d = a\sqrt{3}\). (B) The lateral surface area (LSA) of a cube is the sum of the areas of its four vertical faces. Since each face is a square with area \(a^2\), the LSA is \(4a^2\). Step 1: Find the side length (\(a\)) of the cube using the given diagonal Given: Diagonal of the cube, \(d = \sqrt{6} \text{ cm}\). We know that \(d = a\sqrt{3}\). Substitute the given value of \(d\): \[ \sqrt{6} = a\sqrt{3} \] To solve for \(a\), divide both sides by \(\sqrt{3}\): \[ a = \frac{\sqrt{6}}{\sqrt{3}} \] We can simplify this using the property \(\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}\): \[ a = \sqrt{\frac{6}{3}} \] \[ a = \sqrt{2} \text{ cm} \] So, the side length of the cube is \(\sqrt{2} \text{ cm}\). Step 2: Calculate the lateral surface area (LSA) of the cube The formula for the lateral surface area of a cube is LSA = \(4a^2\). Substitute the value of \(a = \sqrt{2} \text{ cm}\) into this formula: \[ \text{LSA} = 4 (\sqrt{2})^2 \] Since \((\sqrt{2})^2 = 2\), we have: \[ \text{LSA} = 4 \times 2 \] \[ \text{LSA} = 8 \text{ cm}^2 \] Step 3: Compare with the given options The calculated lateral surface area is \(8 \text{ cm}^2\). This matches option (4).