Let's update the solution using the correct length of the cuboid, which is 3 cm instead of 8 cm.

Step 1: Calculate the volume of the cube

The volume of the cube remains:

$V_{\text{cube}} = 6^3 = 216 \, \text{cm}^3$

Step 2: Calculate the height of the cuboid

Given that the volume of the cuboid is equal to the volume of the cube, we have:

$V_{\text{cuboid}} = V_{\text{cube}} = 216 \, \text{cm}^3$

The volume of a cuboid is given by:

$V_{\text{cuboid}} = \text{length} \times \text{breadth} \times \text{height}$

Substitute the correct length:

$216 = 3 \times 4 \times \text{height}$

Solve for the height:

$\text{height} = \frac{216}{3 \times 4} = \frac{216}{12} = 18 \, \text{cm}$

Step 3: Calculate the surface area of the cuboid

The surface area of a cuboid is given by:

$A_{\text{cuboid}} = 2(\text{length} \times \text{breadth} + \text{breadth} \times \text{height} + \text{height} \times \text{length})$

Substitute the known values:

$A_{\text{cuboid}} = 2(3 \times 4 + 4 \times 18 + 18 \times 3)$

Calculate each term:

$A_{\text{cuboid}} = 2(12 + 72 + 54) = 2(138) = 276 \, \text{cm}^2$

Final Answers:

• The height of the cuboid is $18 \, \text{cm}$.
• The surface area of the cuboid is $276 \, \text{cm}^2$.

Would you like any further details or explanations?

Follow-up Questions:

1. How would the height change if the volume of the cube were halved?
2. What would the surface area be if the breadth of the cuboid were reduced to 2 cm?
3. How does changing the dimensions of a cuboid affect its surface area compared to its volume?
4. What is the effect of doubling the length of the cuboid on its surface area?
5. How does the surface area of this cuboid compare to that of the original one with a length of 8 cm?

Tip:

In problems involving volumes and surface areas of solids, always ensure that you substitute the correct dimensions in formulas to avoid calculation errors.