The length of the diagonal of a cube is ` 8 sqrt3 ` cm . Find its
edge
(ii) total surface
(iii) volume

Answer

To solve the problem step by step, we will find the edge length, total surface area, and volume of the cube given that the length of the diagonal is \( 8\sqrt{3} \) cm.

### Step 1: Find the Edge Length of the Cube
The formula for the length of the diagonal \( d \) of a cube in terms of its edge length \( a \) is given by:
\[ d = a\sqrt{3} \]
Given that the diagonal \( d = 8\sqrt{3} \) cm, we can set up the equation:
\[ 8\sqrt{3} = a\sqrt{3} \]
To find \( a \), we can divide both sides by \( \sqrt{3} \):
\[ a = 8 \text{ cm} \]

### Step 2: Calculate the Total Surface Area of the Cube
The formula for the total surface area \( A \) of a cube is:
\[ A = 6a^2 \]
Substituting the value of \( a \):
\[ A = 6 \times (8)^2 \]
Calculating \( 8^2 \):
\[ 8^2 = 64 \]
Now substituting back:
\[ A = 6 \times 64 = 384 \text{ cm}^2 \]

### Step 3: Calculate the Volume of the Cube
The formula for the volume \( V \) of a cube is:
\[ V = a^3 \]
Substituting the value of \( a \):
\[ V = (8)^3 \]
Calculating \( 8^3 \):
\[ 8^3 = 512 \]
So, the volume is:
\[ V = 512 \text{ cm}^3 \]

### Final Answers
1. Edge length of the cube: \( 8 \) cm
2. Total surface area of the cube: \( 384 \) cm\(^2\)
3. Volume of the cube: \( 512 \) cm\(^3\)

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