The edge of a cubical box is 15 cm. A cuboidal box measures 16 cm `times` 12 cm `times` 8 cm. Total surface area of which box is greater ? By how much ?

Answer

To determine which box has a greater total surface area and by how much, we will calculate the total surface area of both the cubical and cuboidal boxes step by step.

### Step 1: Calculate the Total Surface Area of the Cubical Box
The formula for the total surface area (TSA) of a cube is given by:
\[ \text{TSA} = 6 \times \text{side}^2 \]
Given that the edge of the cubical box is 15 cm, we can substitute this value into the formula:
\[ \text{TSA} = 6 \times (15 \, \text{cm})^2 \]
Calculating \( (15 \, \text{cm})^2 \):
\[ (15 \, \text{cm})^2 = 225 \, \text{cm}^2 \]
Now substituting this back into the TSA formula:
\[ \text{TSA} = 6 \times 225 \, \text{cm}^2 = 1350 \, \text{cm}^2 \]

### Step 2: Calculate the Total Surface Area of the Cuboidal Box
The formula for the total surface area of a cuboid is:
\[ \text{TSA} = 2 \times (L \times B + B \times H + H \times L) \]
Where:
- Length \( L = 16 \, \text{cm} \)
- Breadth \( B = 12 \, \text{cm} \)
- Height \( H = 8 \, \text{cm} \)

Substituting the values into the formula:
\[ \text{TSA} = 2 \times (16 \times 12 + 12 \times 8 + 8 \times 16) \]

Calculating each term:
1. \( 16 \times 12 = 192 \)
2. \( 12 \times 8 = 96 \)
3. \( 8 \times 16 = 128 \)

Now adding these values together:
\[ 192 + 96 + 128 = 416 \]

Now substituting this back into the TSA formula:
\[ \text{TSA} = 2 \times 416 = 832 \, \text{cm}^2 \]

### Step 3: Compare the Total Surface Areas
Now we compare the total surface areas of both boxes:
- TSA of Cubical Box = \( 1350 \, \text{cm}^2 \)
- TSA of Cuboidal Box = \( 832 \, \text{cm}^2 \)

### Step 4: Calculate the Difference
To find out by how much the total surface area of the cubical box is greater than that of the cuboidal box, we subtract the TSA of the cuboidal box from that of the cubical box:
\[ \text{Difference} = 1350 \, \text{cm}^2 - 832 \, \text{cm}^2 = 518 \, \text{cm}^2 \]

### Conclusion
The total surface area of the cubical box is greater than that of the cuboidal box by \( 518 \, \text{cm}^2 \).

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