Except for one face of a given cube, identical cubes are glued through their faces to all the other faces of the given cube. If each side of the given cube measures 3 cm, then what is the total surface area of the solid body thus formed?

### Concept & Surface Area Logic When identical cubes are glued to the faces of a central cube, the total surface area changes. Each glued face covers up surface area on both the central cube and the attached cube. $$ \text{Total Exposed Faces} = \text{Exposed Faces on Central Cube} + \text{Exposed Faces on Attached Cubes} $$ ### Step-by-Step Solution * Given: A central cube has identical cubes attached to 5 of its 6 faces (since one face is excepted). * Total cubes = 1 central + 5 attached = 6 cubes. * Let's count the exposed faces on the central cube: Since 5 faces are glued, only 1 face remains exposed. * Let's count the exposed faces on the attached cubes: Each of the 5 attached cubes has 1 face glued to the central cube, leaving 5 faces exposed per attached cube. * Total exposed faces = $1 + (5 \times 5) = 1 + 25 = 26$ faces. * The side length of each cube is $3\text{ cm}$. Area of one face = $3 \times 3 = 9\text{ cm}^2$. * Total surface area = $26 \times 9 = 234\text{ cm}^2$. ### Exam Strategy & Shortcut Instead of calculating the total initial area and subtracting hidden faces, directly count the exposed faces of the final 3D figure. 1 central face + $5 \times 5$ attached faces = 26 faces. Multiply by face area (9) to get 234 instantly. ### Common Pitfall A common mistake is assuming all 6 faces of the central cube have attached cubes, or miscalculating the covered faces by subtracting incorrectly from the aggregate area. ### Final Answer Therefore, the correct answer is **234 cm²**.