The volume of a cube is `729cm^(3)`. Find its total surface area

Answer

To find the total surface area of a cube when given its volume, we can follow these steps:

### Step 1: Understand the formula for the volume of a cube.
The volume \( V \) of a cube is given by the formula:
\[ V = A^3 \]
where \( A \) is the length of one side of the cube.

### Step 2: Set up the equation using the given volume.
We know from the problem that the volume of the cube is \( 729 \, \text{cm}^3 \). Therefore, we can write:
\[ A^3 = 729 \]

### Step 3: Solve for \( A \).
To find \( A \), we need to take the cube root of both sides:
\[ A = \sqrt[3]{729} \]
Calculating the cube root, we find:
\[ A = 9 \, \text{cm} \]

### Step 4: Use the side length to find the total surface area.
The total surface area \( TSA \) of a cube is given by the formula:
\[ TSA = 6A^2 \]
Now, substituting the value of \( A \):
\[ TSA = 6 \times (9)^2 \]

### Step 5: Calculate \( (9)^2 \).
Calculating \( (9)^2 \):
\[ (9)^2 = 81 \]

### Step 6: Substitute back to find TSA.
Now substituting back into the TSA formula:
\[ TSA = 6 \times 81 \]

### Step 7: Calculate the total surface area.
Calculating \( 6 \times 81 \):
\[ TSA = 486 \, \text{cm}^2 \]

Thus, the total surface area of the cube is \( 486 \, \text{cm}^2 \).

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