Surface Area of 3D Shapes - Solution & Explanation

Problem Overview: You are given an m x n grid where grid[i][j] represents the number of unit cubes stacked at that position. The task is to compute the total surface area of the resulting 3D structure. Adjacent cubes share faces, so internal faces must not be counted.

The grid acts like a height map. Each stack forms a vertical column of cubes. A single cube contributes 6 faces, but whenever two cubes touch, the touching faces disappear from the external surface. The goal is to count only the faces visible from the outside.

Approach 1: Direct Calculation of Surface Area (O(m*n) time, O(1) space)

Iterate through every cell of the grid and treat each stack independently. A stack with height h contributes 4*h + 2 surface area if it stands alone: four vertical sides plus top and bottom. The key adjustment comes from neighboring stacks. If the current stack touches another stack horizontally, the touching faces are hidden. Subtract 2 * min(currentHeight, neighborHeight) for each adjacent neighbor (typically checking top and left neighbors avoids double counting). This method works well because it directly models how shared faces reduce the exposed surface. The algorithm scans the grid once, performing constant-time arithmetic per cell.

This approach relies on simple iteration over a matrix while applying arithmetic rules derived from geometry. It avoids building any additional structures and keeps the logic compact.

Approach 2: Iterative Side-by-Side Calculation (O(m*n) time, O(1) space)

Another way to think about the problem is to count visible faces from each direction instead of starting with a full formula. Iterate through the grid and add the top and bottom surfaces whenever grid[i][j] > 0. Then compute the exposed sides by comparing the current stack with its neighbors. For example, the visible surface on the left side equals max(0, grid[i][j] - grid[i][j-1]). Repeat this for right, front, and back directions. Each comparison measures how much of the current stack extends beyond its neighbor.

This perspective mirrors how you would visually inspect the shape layer by layer. It’s still a simple scan over a array-based grid, but instead of subtracting overlaps, it adds only the exposed differences between adjacent stacks.

Recommended for interviews: The direct calculation approach is usually what interviewers expect. It demonstrates that you recognize shared faces between neighboring stacks and can translate the geometric observation into code. The side-by-side comparison method is equally efficient but emphasizes reasoning about exposed edges. Showing either approach with O(m*n) time and O(1) space signals solid understanding of grid traversal and surface counting.