Calculating the Volume of a Cuboid Formed by Joining Cubes

This problem involves understanding how joining two cubes changes their dimensions to form a new shape, a cuboid, and then calculating the volume of this resulting cuboid.

We are given two identical cubes, each with an edge length of 14 cm.

When two such cubes are joined together along one face, the resulting shape is a cuboid. The dimensions of this new cuboid will be different from the original cubes.

• Let the edge length of the original cubes be \(a = 14\) cm.
• When two cubes are joined face-to-face, one dimension (let's call it length) is doubled, while the other two dimensions (width and height) remain the same as the edge of the original cube.

So, the dimensions of the cuboid formed are:

• Length (L) = Edge length of cube + Edge length of cube = \(a + a = 2a\)
• Width (W) = Edge length of cube = \(a\)
• Height (H) = Edge length of cube = \(a\)

Substituting the given value \(a = 14\) cm:

• Length (L) = \(2 \times 14\) cm = 28 cm
• Width (W) = 14 cm
• Height (H) = 14 cm

Now we need to calculate the volume of this cuboid. The formula for the volume of a cuboid is:

\( \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \)

Plugging in the dimensions of the formed cuboid:

\( \text{Volume} = L \times W \times H \)

\( \text{Volume} = 28 \text{ cm} \times 14 \text{ cm} \times 14 \text{ cm} \)

Let's perform the calculation step-by-step:

\( 14 \times 14 = 196 \)

So, \( \text{Volume} = 28 \times 196 \text{ cm}^3 \)

Now, multiply 28 by 196:

\( 28 \times 196 = 5488 \)

Therefore, the volume of the cuboid formed is 5488 cm\(^3\).

Let's check this calculation:

The volume of the cuboid formed by joining the two cubes is 5488 cm\(^3\).

Revision Table: Cube and Cuboid Volume Calculation

Notice that the volume of the cuboid (5488 cm\(^3\)) is exactly twice the volume of a single cube (2744 cm\(^3\)), which makes sense because the cuboid is formed by joining two identical cubes.

Additional Information on Solid Shapes and Volume

Understanding basic solid shapes like cubes and cuboids is fundamental in geometry. The volume of a solid measures the amount of three-dimensional space it occupies.

• Cube: A cube is a special type of cuboid where all its edges are equal in length. It has 6 square faces, 12 edges, and 8 vertices. Its symmetry makes volume calculation straightforward using the formula \(a^3\), where \(a\) is the edge length.
• Cuboid: A cuboid, also known as a rectangular prism, is a solid with six rectangular faces at right angles to each other. It has 12 edges and 8 vertices. Its volume is calculated by multiplying its three distinct dimensions: length, width, and height.
• Joining Solids: When solids are joined, their dimensions change, forming a new combined solid. The volume of the new solid is the sum of the volumes of the original solids, provided there is no overlap. In this case, joining two cubes of volume \(V_{\text{cube}}\) results in a cuboid with volume \(V_{\text{cuboid}} = V_{\text{cube}} + V_{\text{cube}} = 2 \times V_{\text{cube}}\). Our calculation confirms this: \(2 \times 2744 \text{ cm}^3 = 5488 \text{ cm}^3\).
• Units: Volume is measured in cubic units, such as cm\(^3\) or m\(^3\). This indicates that we are measuring a three-dimensional space.

Calculating the volume of composite shapes often involves breaking down the composite shape into simpler, known shapes like cubes and cuboids, calculating their individual volumes, and then adding them up.