Finding the Surface Area of the Resulting Cuboid

When two cubes, each having an edge of length 12 cm, are joined end to end, they form a single cuboid. Understanding how the dimensions change is key to calculating the surface area of the resulting cuboid.

Let the edge length of each cube be \(a\). Given \(a = 12 \, \text{cm}\).

When two such cubes are joined end to end, the length of the resulting shape becomes the sum of the lengths of the two edges along the direction they are joined. The other two dimensions (breadth and height) remain the same as the edge length of a single cube.

• Length of the resulting cuboid, \(l = a + a = 2a\)
• Breadth of the resulting cuboid, \(b = a\)
• Height of the resulting cuboid, \(h = a\)

Substituting the given value \(a = 12 \, \text{cm}\):

• Length, \(l = 2 \times 12 \, \text{cm} = 24 \, \text{cm}\)
• Breadth, \(b = 12 \, \text{cm}\)
• Height, \(h = 12 \, \text{cm}\)

The formula for the total surface area of a cuboid with length \(l\), breadth \(b\), and height \(h\) is given by:

\(\text{Surface Area} = 2(lb + bh + hl)\)

Now, we substitute the dimensions of the resulting cuboid into this formula:

\(\text{Surface Area} = 2((24 \, \text{cm})(12 \, \text{cm}) + (12 \, \text{cm})(12 \, \text{cm}) + (12 \, \text{cm})(24 \, \text{cm}))\)

Let's calculate the terms inside the parenthesis:

• \(lb = 24 \times 12 = 288\)
• \(bh = 12 \times 12 = 144\)
• \(hl = 12 \times 24 = 288\)

Summing these values:

\(lb + bh + hl = 288 + 144 + 288 = 720\)

Now, multiply by 2 to get the total surface area:

\(\text{Surface Area} = 2 \times 720 \, \text{cm}^2 = 1440 \, \text{cm}^2\)

Thus, the surface area of the resulting cuboid is 1440 cm².