Three cubes each of volume 343 cm³ are placed side by side. What will be the surface area of the solid so formed (in cm²)?
686
Calculating Surface Area of Combined Cubes
This problem involves finding the surface area of a new solid formed by joining three identical cubes side by side. First, we need to determine the dimensions of a single cube and then understand how placing them together changes the overall shape and its exposed surface area.
Step 1: Find the Side Length of Each Cube
The volume of a cube is given by the formula \(V = s^3\), where \(s\) is the side length. We are given that the volume of each cube is 343 cm³.
So, we have:
\(s^3 = 343 \text{ cm}^3\)
To find the side length \(s\), we need to calculate the cube root of 343.
\(s = \sqrt[3]{343}\)
We know that \(7 \times 7 \times 7 = 49 \times 7 = 343\). Therefore:
\(s = 7 \text{ cm}\)
The side length of each cube is 7 cm.
Step 2: Determine the Dimensions of the Solid Formed
When three identical cubes are placed side by side, they form a rectangular prism. Let's assume they are placed along the length axis.
The dimensions of the resulting solid will be:
- Length (L): The sum of the side lengths of the three cubes along the joined axis. \(L = s + s + s = 3s\)
- Width (W): The side length of one cube. \(W = s\)
- Height (H): The side length of one cube. \(H = s\)
Substituting the value of \(s = 7\) cm:
- Length \(L = 3 \times 7 = 21 \text{ cm}\)
- Width \(W = 7 \text{ cm}\)
- Height \(H = 7 \text{ cm}\)
| Dimension | Value (cm) |
|---|---|
| Length (L) | 21 |
| Width (W) | 7 |
| Height (H) | 7 |
Step 3: Calculate the Surface Area of the Formed Solid
The solid formed is a rectangular prism. The surface area of a rectangular prism is given by the formula:
\(SA = 2(LW + LH + WH)\)
Substitute the dimensions we found:
\(SA = 2((21 \text{ cm} \times 7 \text{ cm}) + (21 \text{ cm} \times 7 \text{ cm}) + (7 \text{ cm} \times 7 \text{ cm}))\)
Calculate the area of each pair of faces:
- Area of two faces with dimensions L and W: \(2 \times (21 \times 7) = 2 \times 147 = 294 \text{ cm}^2\)
- Area of two faces with dimensions L and H: \(2 \times (21 \times 7) = 2 \times 147 = 294 \text{ cm}^2\)
- Area of two faces with dimensions W and H: \(2 \times (7 \times 7) = 2 \times 49 = 98 \text{ cm}^2\)
Summing these areas to find the total surface area:
\(SA = 294 \text{ cm}^2 + 294 \text{ cm}^2 + 98 \text{ cm}^2\)
\(SA = 588 \text{ cm}^2 + 98 \text{ cm}^2\)
\(SA = 686 \text{ cm}^2\)
Alternatively, using the formula directly:
\(SA = 2((21 \times 7) + (21 \times 7) + (7 \times 7))\)
\(SA = 2(147 + 147 + 49)\)
\(SA = 2(294 + 49)\)
\(SA = 2(343)\)
\(SA = 686 \text{ cm}^2\)
The surface area of the solid formed is 686 cm².
Understanding Surface Area of Combined Cubes
When cubes are joined, some faces become internal and are no longer part of the external surface area. A single cube has 6 faces. Three separate cubes have \(3 \times 6 = 18\) faces in total.
When placed side by side in a line, the first cube joins the second, and the second joins the third. For each join, two faces (one from each cube) are covered. Since there are two joins, \(2 \times 2 = 4\) faces are covered.
The number of exposed faces is \(18 - 4 = 14\) faces.
Each face is a square with area \(s^2 = 7^2 = 49 \text{ cm}^2\).
Total surface area = Number of exposed faces \(\times\) Area of one face
Total surface area = \(14 \times 49 \text{ cm}^2\)
Total surface area = \(686 \text{ cm}^2\)
Both methods yield the same result, confirming the surface area calculation.
Revision Table: Geometry Formulas
| Shape | Formula for Volume (V) | Formula for Surface Area (SA) |
|---|---|---|
| Cube (side s) | \(V = s^3\) | \(SA = 6s^2\) |
| Rectangular Prism (l, w, h) | \(V = lwh\) | \(SA = 2(lw + lh + wh)\) |
Additional Information: Combining Solids
When combining geometric solids, the volume of the new solid is typically the sum of the volumes of the original solids, assuming no overlap. In this case, the volume of the new solid is \(3 \times 343 = 1029 \text{ cm}^3\), which is \(21 \text{ cm} \times 7 \text{ cm} \times 7 \text{ cm}\).
However, the surface area calculation is different. The surface area is the total area of the exposed faces of the combined solid. When solids are joined, the faces that are in contact with each other are no longer part of the surface area. Therefore, to find the surface area of the combined solid, you calculate the total surface area of the individual solids and subtract the area of the faces that are joined.
In our problem:
- Total surface area of three separate cubes = \(3 \times (6s^2) = 18s^2\).
- Area of each joined face = \(s^2\).
- Number of joined faces = 4 (2 pairs).
- Area of joined faces = \(4s^2\).
- Surface area of combined solid = \(18s^2 - 4s^2 = 14s^2\).
Using \(s = 7\) cm:
\(SA = 14 \times (7 \text{ cm})^2 = 14 \times 49 \text{ cm}^2 = 686 \text{ cm}^2\).
This confirms that the surface area of the resulting solid is the total area of 14 faces, each with area \(s^2\).
Important Questions from Volume and Surface Area
If the base radius of a cone is doubled and its height is halved, then the volume of the new cone will be:
How many solid spherical balls, each of diameter 1.5 cm, can be made by melting a solid cylinder with height 36cm and base radius 8cm?
A solid metallic sphere of radius 8 cm is melted and recasted as a cone of height 8 cm. Find the base radius of the cone (in cm).
The volume of a wall which is 5 times as high as it is broad and 8 times as long as it is high, is 12.8m³. The breadth of the wall is:
The volumes of 3 solid cubes made of metal are 125 cm3, 64 cm3 and 27 cm3 respectively. After melting all the three cubes a solid cube is made. Find the edge of the new cube.
