This problem asks us to find the total surface area of a cuboid formed by joining six identical cubes end to end. Each cube has an edge length of 2 cm. When six cubes are joined end to end, they form a rectangular prism, which is a cuboid.

Understanding the Dimensions of the Resulting Cuboid

When six cubes of side length 2 cm are joined end to end, the dimensions of the resulting cuboid change in only one direction: the length. The width and height remain the same as the side length of a single cube.

• The length of the cuboid will be the sum of the edge lengths of the six cubes along the direction they are joined. So, Length = 6 $\times$ 2 cm = 12 cm.
• The width of the cuboid will be equal to the edge length of a single cube. So, Width = 2 cm.
• The height of the cuboid will be equal to the edge length of a single cube. So, Height = 2 cm.

So, the dimensions of the resulting cuboid are 12 cm $\times$ 2 cm $\times$ 2 cm.

Calculating the Total Surface Area of the Cuboid

The formula for the total surface area (TSA) of a cuboid with length \(l\), width \(w\), and height \(h\) is:

$$\text{TSA} = 2(lw + lh + wh)$$

Using the dimensions of our cuboid (\(l = 12\) cm, \(w = 2\) cm, \(h = 2\) cm), we can calculate the total surface area:

$$\text{TSA} = 2((12 \text{ cm} \times 2 \text{ cm}) + (12 \text{ cm} \times 2 \text{ cm}) + (2 \text{ cm} \times 2 \text{ cm}))$$

$$\text{TSA} = 2((24 \text{ cm}^2) + (24 \text{ cm}^2) + (4 \text{ cm}^2))$$

$$\text{TSA} = 2(24 + 24 + 4) \text{ cm}^2$$

$$\text{TSA} = 2(52) \text{ cm}^2$$

$$\text{TSA} = 104 \text{ cm}^2$$

The total surface area of the resulting cuboid is 104 cm$^2$.

Understanding Surface Area Changes

When cubes are joined, some faces are hidden inside the resulting solid and no longer contribute to the total surface area. For each point where two cubes are joined, two faces (one from each cube) become internal. In this case, six cubes joined end-to-end have 5 joints between them. Each joint hides two faces. So, $5 \times 2 = 10$ faces are hidden. Each cube initially has 6 faces. Six cubes have a total of $6 \times 6 = 36$ faces. If 10 faces are hidden, the remaining number of faces contributing to the surface area would seem to be $36 - 10 = 26$. Each face has an area of $(2 \text{ cm})^2 = 4 \text{ cm}^2$. So, the total surface area is $26 \times 4 \text{ cm}^2 = 104 \text{ cm}^2$. This confirms the cuboid method calculation.

Revision Table: Key Concepts for Surface Area Calculation

Additional Information: Solids and Mensuration

Mensuration is the branch of mathematics that deals with the measurement of lengths, areas, and volumes of geometric figures. Solid shapes, like cubes and cuboids, are three-dimensional figures that occupy space.

• Cube: A cube is a special type of cuboid where all edges are of equal length, and all faces are squares. It has 6 faces, 12 edges, and 8 vertices.
• Cuboid: A cuboid is a three-dimensional shape with six rectangular faces at right angles to each other. It has 6 faces, 12 edges, and 8 vertices. A cube is a specific type of cuboid.
• Surface Area: The total surface area of a solid is the sum of the areas of all its faces. It is measured in square units (e.g., cm$^2$, m$^2$).
• Volume: The volume of a solid is the amount of space it occupies. It is measured in cubic units (e.g., cm$^3$, m$^3$). For a cuboid, Volume = \(l \times w \times h\). For a cube, Volume = \(s^3\).

Understanding how joining or cutting solids affects their surface area is important in mensuration problems. The total surface area of a composite solid is the sum of the exposed surface areas of its parts.