9.5. Volume of Cube, Cuboid and Cylinder

Amount of space occupied by a three dimensional object is called its volume. Try to compare the volume of objects surrounding you. For example, volume of a room is greater than the volume of an almirah kept inside it. Similarly, volume of your pencil box is greater than the volume of the pen and the eraser kept inside it.

Volume measures how much space an object takes up. When comparing different objects, we can see that some have more space than others. Imagine you have a big room (like a classroom) and inside it, you have a small almirah (a type of cupboard). The room holds much more air and space compared to the almirah. Similarly, within a pencil box, the space is more than that of the pen and eraser inside it because the box can hold more items. Understanding this helps us visualize the concept of volume as a measurement of space.

Think of volume like packing boxes for a moving day. A big box can fit more items than a smaller one, just like the room can hold more than the almirah. When we say something has a larger volume, we can think of it as being able to fit more things inside.

Remember, we use square units to find the area of a region. Here we will use cubic units to find the volume of a solid, as cube is the most convenient solid shape (just as square is the most convenient shape to measure area of a region). For finding the area we divide the region into square units, similarly, to find the volume of a solid we need to divide it into cubical units.

To measure area, we use square units (like square meters or square centimeters) because they represent two-dimensional space. To find volume, which is three-dimensional, we use cubic units (like cubic meters or cubic centimeters). For example, if we have a cube that is 1 cm on each side, it can fit in exactly 1 cubic centimeter of space. We visualize volume by imagining filling a container with cube-like blocks, just as we would cover a surface with squares for area.

Imagine you are filling a swimming pool with water. The amount of water needed to fill the pool can be thought of in cubic terms, just like how measuring the grass needed for a lawn would be in square terms. Each cubic block of water takes up volume, making it the best way to gauge how much water is required.

Since we have used 36 cubes to form these cuboids, volume of each cuboid is 36 cubic units. Also, volume of each cuboid is equal to the product of length, breadth and height of the cuboid. From the above example we can say volume of cuboid = l × b × h. Since l × b is the area of its base we can also say that, Volume of cuboid = area of the base × height.

A cuboid is defined by its three dimensions: length, breadth, and height. To find the volume, you simply multiply these dimensions together (the formula is Volume = l × b × h). If you construct a cuboid using smaller cubes, like stacking 36 1-inch cubes together, the total volume, 36 cubic inches, represents how many of the smaller cubical units fit into the larger cuboid. The area of the base times the height will give you the same volume because it counts how many slices of the base fit into the height as well.

Consider a shipping container. You want to know how many boxes you can fit inside. If the container is 10 meters long, 5 meters wide, and 4 meters tall, multiplying these dimensions (10 × 5 × 4) gives you the total number of cubic meters of space available for your boxes.

The cube is a special case of a cuboid, where l = b = h. Hence, volume of cube = l × l × l = l³.

A cube is a specific type of cuboid where all sides are equal. Therefore, the formula for calculating its volume is a simple extension of the cuboid formula: Volume = l × l × l or l³. If each side of the cube is 3 cm, the volume is 3 × 3 × 3 = 27 cubic cm. This shows how the same principle applies to cubes, emphasizing their uniform structure.

Think about a sugar cube. If you have a cube where each side is 1 cm, the total amount of sugar in that single cube can be visualized as occupying a certain volume. If you stack several cubes, you can easily calculate the total volume using the cube formula.

Volume of cylinder = area of base × height = πr² × h.

To find the volume of a cylinder, you again use the concept of area and height. The base of a cylinder is a circle, so the area formula is πr² (where r is the radius). By multiplying this area by the height of the cylinder, you get the total volume: Volume = πr² × h. This is because you're stacking those circles up to create the cylinder, just like stacking layers of flat disks.

Consider a soda can. The amount of soda it holds can be calculated using the cylinder volume formula. The base circle is where the liquid is filled (the lid of the can), and the height is the depth of the can, giving you the total volume of soda it can carry.

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