Understanding the Problem: Surface Area of Joined Cubes

The question asks us to find the surface area of a new shape formed by joining three identical cubes end-to-end. We are given the diagonal length of a single cube. First, we need to use the diagonal length to find the side length of the cube. Once we have the side length, we can determine the dimensions of the resulting cuboid and then calculate its surface area.

Step 1: Find the Side Length of the Cube

The diagonal of a cube is related to its side length by a specific formula. Let 'a' be the side length of the cube. The formula for the diagonal (d) of a cube is:

$$d = a\sqrt{3}$$

We are given that the diagonal of the cube is $6\sqrt{3}$ cm.

So, we have:

$$a\sqrt{3} = 6\sqrt{3} \text{ cm}$$

To find 'a', we can divide both sides by $\sqrt{3}$:

$$a = \frac{6\sqrt{3}}{\sqrt{3}} \text{ cm}$$

$$a = 6 \text{ cm}$$

Thus, the side length of each of the three equal volume cubes is 6 cm.

Step 2: Determine the Dimensions of the Resulting Cuboid

When three identical cubes of side length 'a' are joined end-to-end, they form a cuboid. Let's determine the length (L), width (W), and height (H) of this new cuboid.

• The length (L) will be the sum of the side lengths of the three cubes along the direction they are joined. So, $L = a + a + a = 3a$.
• The width (W) remains the side length of a single cube, as they are joined along one dimension. So, $W = a$.
• The height (H) also remains the side length of a single cube. So, $H = a$.

Using the side length $a = 6$ cm, the dimensions of the cuboid are:

• Length (L) = $3 \times 6 = 18$ cm
• Width (W) = 6 cm
• Height (H) = 6 cm

We can summarize the dimensions in a table:

Step 3: Calculate the Surface Area of the Cuboid

The formula for the total surface area (SA) of a cuboid with length L, width W, and height H is:

$$SA = 2(LW + LH + WH)$$

Now, we substitute the dimensions of our cuboid (L=18 cm, W=6 cm, H=6 cm) into the formula:

$$SA = 2 \times ( (18 \times 6) + (18 \times 6) + (6 \times 6) ) \text{ cm}^2$$

Let's calculate the products inside the parentheses:

• $LW = 18 \times 6 = 108$
• $LH = 18 \times 6 = 108$
• $WH = 6 \times 6 = 36$

Now substitute these values back into the surface area formula:

$$SA = 2 \times (108 + 108 + 36) \text{ cm}^2$$

Add the numbers inside the parentheses:

$$108 + 108 + 36 = 216 + 36 = 252$$

Finally, multiply by 2:

$$SA = 2 \times 252 \text{ cm}^2$$

$$SA = 504 \text{ cm}^2$$

The surface area of the resulting cuboid is 504 cm$^2$.

Summary of Calculations

We started with the diagonal of a cube, found the side length, used the side length to get the cuboid dimensions, and finally calculated the surface area of the cuboid. Each step is crucial in solving this geometry problem involving the surface area of joined cubes.

Revision Table: Key Formulas

Additional Information: Solids and Surface Area

This problem combines concepts of cubes and cuboids, which are types of polyhedra. Understanding how shapes transform when joined is important in geometry. The surface area calculation is key in many practical applications, such as determining the amount of material needed to cover an object. When cubes are joined, some faces become internal and are no longer part of the total surface area, which is why the surface area of the resulting cuboid is not simply the sum of the surface areas of the three individual cubes (which would be $3 \times 6a^2 = 18a^2 = 18 \times 6^2 = 18 \times 36 = 648$ cm$^2$). The reduction in surface area is due to the faces that are glued together.