Given information:
- The sum of length, breadth, and height of a cuboidal room is 26m.
- The length of the diagonal is 20m.

To find:
- The total surface area of the cuboid.

Step 1: Understanding the problem:
A cuboid is a three-dimensional figure with six rectangular faces. The total surface area of a cuboid is the sum of the areas of all its faces.

Step 2: Identifying the parameters:
Let's assume the length of the cuboid as L, breadth as B, and height as H.

Step 3: Formulating the equations:
We are given that the sum of length, breadth, and height is 26m:
L + B + H = 26 ...(Equation 1)

We are also given that the length of the diagonal is 20m. Using the Pythagorean theorem, we can find the relationship between the length, breadth, and height:
L² + B² + H² = (diagonal)² = 20² = 400 ...(Equation 2)

Step 4: Solving the equations:
We have two equations (Equation 1 and Equation 2) and three variables (L, B, and H). To solve this system of equations, we can use substitution or elimination method.

From Equation 1, we can express L as:
L = 26 - B - H

Substituting this value of L in Equation 2:
(26 - B - H)² + B² + H² = 400
Simplifying this equation will give us a quadratic equation in terms of B and H.

Step 5: Finding the values of L, B, and H:
Solving the quadratic equation will give us two sets of values for B and H. We need to consider the positive values since dimensions cannot be negative.

Step 6: Calculating the total surface area:
The total surface area of the cuboid is the sum of the areas of all its faces. The formula for the surface area of a cuboid is:
Surface Area = 2(LB + BH + HL)

Using the values of L, B, and H, we can substitute them into the formula to calculate the total surface area.

Step 7: Final answer:
The total surface area of the cuboid is the calculated value from Step 6.

This is math questions tell