Making a cube from cuboids involves combining multiple cuboid pieces to form a larger shape where all sides are equal in length. This is possible because a cube is a special form of cuboid in which length, breadth and height are equal, meaning all dimensions of a cube are equal in measure.
Understanding the Shapes: Cube vs. Cuboid
- Cuboid: A three-dimensional solid shape with six rectangular faces at right angles to each other. It has a length, breadth (or width), and height.
- Cube: A special type of cuboid where all six faces are squares of the same size. Consequently, its length, breadth, and height are all equal.
The Concept of Forming a Cube from Smaller Cuboids
To construct a larger cube from smaller, identical cuboids, you must arrange and stack the cuboids in such a way that the overall resulting shape has equal dimensions in length, breadth, and height.
The key principle is that the side length of the resulting larger cube must be a common multiple of the length, breadth, and height of the individual cuboids. The smallest possible cube you can form will have a side length equal to the Least Common Multiple (LCM) of the cuboid's dimensions.
Calculating the Number of Cuboids Needed
Once the dimensions of the smallest possible cube are determined (using the LCM), you can calculate how many individual cuboids are required. This is typically done by comparing the total volume of the desired cube to the volume of a single cuboid.
Number of Cuboids = (Volume of the formed Cube) / (Volume of a single Cuboid)
Volume of a Cube = side side side (or side³)
Volume of a Cuboid = length breadth height
Example: Using 5 cm x 2 cm x 5 cm Cuboids
Let's use the example provided in the reference to illustrate this process:
- Individual Cuboid Dimensions:
- Length = 5 cm
- Breadth = 2 cm
- Height = 5 cm
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Find the side of the smallest possible cube: The side length of the cube must be a multiple of 5 cm, 2 cm, and 5 cm. The smallest such number is the LCM of 5, 2, and 5.
- LCM(5, 2, 5) = LCM(5, 2) = 10 cm.
- So, the smallest cube formed would have a side length of 10 cm.
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Calculate the volume of the cube:
- Volume = 10 cm 10 cm 10 cm = 1000 cubic cm.
-
Calculate the volume of a single cuboid:
- Volume = 5 cm 2 cm 5 cm = 50 cubic cm.
-
Determine the number of cuboids needed:
- Number of Cuboids = Volume of Cube / Volume of Cuboid
- Number of Cuboids = 1000 cm³ / 50 cm³ = 20.
Based on this calculation and the provided information: Therefore, 20 cuboids each of sides 5 cm, 2 cm, 5 cm will be needed to form a cube.
This table summarizes the example:
| Feature | Cuboid (single) | Cube (formed) |
|---|---|---|
| Dimensions | 5 cm x 2 cm x 5 cm | 10 cm x 10 cm x 10 cm |
| Volume | 50 cm³ | 1000 cm³ |
| Number Needed | N/A | 20 |
Practical Assembly
Physically making the cube involves arranging the 20 cuboids. For a 10x10x10 cm cube using 5x2x5 cm cuboids:
- Along the 10 cm length, you can fit 10 cm / 5 cm = 2 cuboids.
- Along the 10 cm breadth, you can fit 10 cm / 2 cm = 5 cuboids.
- Along the 10 cm height, you can fit 10 cm / 5 cm = 2 cuboids.
So, you'd stack them in a 2 x 5 x 2 arrangement of cuboids, which gives 2 5 2 = 20 cuboids in total.
In summary, making a cube from identical cuboids is achieved by arranging them into a larger shape with equal dimensions, calculating the required number based on the volumes and dimensions of the shapes involved.
