Let the dimensions of the cuboid be represented by:

• Length = $l$
• Breadth = $b$
• Height = $h$

The sum of the areas of the six faces of a cuboid is given by the formula:

$2(lb + bh + hl)$

We are told that the sum of the areas is 322 cm², so:

$2(lb + bh + hl) = 322$

Dividing both sides by 2:

$lb + bh + hl = 161$

Next, the dimensions are in the ratio 1:2:3, so we can express $l$, $b$, and $h$ in terms of a common variable $x$:

$l = x, \quad b = 2x, \quad h = 3x$

Substituting these into the equation $lb + bh + hl = 161$:

$x(2x) + 2x(3x) + 3x(x) = 161$

Simplifying:

$2x^2 + 6x^2 + 3x^2 = 161$

$11x^2 = 161$

Solving for $x^2$:

$x^2 = \frac{161}{11} = 14.636$

Taking the square root of both sides:

$x \approx \sqrt{14.636} \approx 3.83$

Now that we have $x \approx 3.83$, we can find the dimensions:

• Length $l = x \approx 3.83$ cm
• Breadth $b = 2x \approx 2 \times 3.83 \approx 7.66$ cm
• Height $h = 3x \approx 3 \times 3.83 \approx 11.49$ cm

Thus, the dimensions of the cuboid are approximately:

• Length = 3.83 cm
• Breadth = 7.66 cm
• Height = 11.49 cm

Would you like further details or clarifications? Here's a set of related questions:

1. How does the surface area formula for a cuboid derive from its face areas?
2. How can the ratio of dimensions affect the calculation of surface area?
3. What is the significance of the value $x$ in the context of cuboid dimensions?
4. How can we solve this problem if the dimensions were in a different ratio, say 2:3:4?
5. How would the calculation change if the cuboid's volume was also given?

Tip: When dealing with ratios in geometry, always express the variables in terms of a common factor before proceeding with the equation to simplify the process.