Given:
The sum of the length, breadth, and depth of a cuboid is 19 cm.
The length of its diagonal is 11 cm.

To find:
The surface area of the cuboid.

Solution:

Step 1: Identifying the dimensions of the cuboid
Let's assume the length, breadth, and depth of the cuboid as l, b, and d respectively.

Step 2: Formulating the equations
From the given information, we can form two equations:
Equation 1: l + b + d = 19
Equation 2: Diagonal (d) = 11 cm

Step 3: Solving the equations
Let's solve the equations to find the values of l, b, and d.
From Equation 1, we can rewrite it as:
l = 19 - b - d

Substituting the value of l in Equation 2, we get:
(19 - b - d)^2 + b^2 + d^2 = 11^2

Simplifying the equation, we get:
b^2 + d^2 + 19^2 - 2 * 19 * b - 2 * 19 * d + 2 * b * d = 121

Rearranging the terms, we get:
b^2 + d^2 - 2 * 19 * b - 2 * 19 * d + 2 * b * d = 121 - 19^2

Step 4: Applying the formula for surface area
The surface area of a cuboid is given by the formula:
Surface Area = 2(lb + bh + hl)

Substituting the values of l, b, and h, we get:
Surface Area = 2[(19 - b - d)(b + d) + b(d) + (19 - b - d)(d)]

Simplifying the equation, we get:
Surface Area = 2[19b + 19d - b^2 - d^2]

Step 5: Calculating the surface area
Now, we have the equation for the surface area in terms of b and d. We can substitute the values of b and d obtained from the previous equation and calculate the surface area.

Conclusion:
By following the above steps, we can find the surface area of the cuboid once we have the values of b and d.