Four identical cubes are joined end to end to form a cuboid .If the total surface area of the resulting cuboid is 648 `cm^(2)` .Find the length of edge of each cube.

Answer

To solve the problem of finding the length of the edge of each cube when four identical cubes are joined end to end to form a cuboid with a total surface area of 648 cm², we can follow these steps:

### Step 1: Understand the dimensions of the cuboid
When four identical cubes are joined end to end, the dimensions of the resulting cuboid can be defined as follows:
- Length (L) = 4a (since there are 4 cubes, each with edge length 'a')
- Breadth (B) = a (the breadth of the cuboid is the same as the edge of a cube)
- Height (H) = a (the height of the cuboid is also the same as the edge of a cube)

### Step 2: Write the formula for the total surface area of the cuboid
The formula for the total surface area (TSA) of a cuboid is given by:
\[ \text{TSA} = 2(LB + BH + LH) \]
Substituting the dimensions we defined:
\[ \text{TSA} = 2(4a \cdot a + a \cdot a + 4a \cdot a) \]

### Step 3: Simplify the expression for TSA
Now, simplify the expression:
\[ \text{TSA} = 2(4a^2 + a^2 + 4a^2) \]
\[ \text{TSA} = 2(9a^2) \]
\[ \text{TSA} = 18a^2 \]

### Step 4: Set the TSA equal to the given value
We know from the problem that the total surface area is 648 cm²:
\[ 18a^2 = 648 \]

### Step 5: Solve for \( a^2 \)
To find \( a^2 \), divide both sides by 18:
\[ a^2 = \frac{648}{18} \]
\[ a^2 = 36 \]

### Step 6: Solve for \( a \)
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{36} \]
\[ a = 6 \, \text{cm} \]

### Conclusion
The length of the edge of each cube is 6 cm.

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