Mensuration - Volume of Cube, Cuboid, and Cylinder

Volume is defined as the total three-dimensional space occupied by a solid object. It is measured in cubic units such as $cm^3$ or $m^3$. Visually, you can imagine volume as the number of unit cubes that can fit perfectly inside a container without overlapping.

A Cube is a regular solid with six equal square faces. Every edge has the same length, denoted as $a$. Visually, a cube looks like a perfectly symmetrical die where the length, breadth, and height are all equal, creating a uniform 3D shape.

A Cuboid is a solid with six rectangular faces. It is defined by three dimensions: length ($l$), breadth ($b$), and height ($h$). Visually, it resembles a shoebox or a brick where the perpendicular edges meet at right angles, but the sides are not necessarily equal.

A Cylinder consists of two identical circular bases connected by a curved surface. The dimensions involved are the radius ($r$) of the circular base and the vertical height ($h$). Visually, it looks like a soda can or a section of a pipe, where the cross-section remains a constant circle throughout its height.

The general principle for finding the volume of right prisms (like cubes, cuboids, and cylinders) is multiplying the Area of the Base by the Height. For a cylinder, the circular base area is $\pi r^2$, and for a cuboid, the rectangular base area is $l \times b$.

Capacity refers to the volume of liquid or gas a hollow object can hold. The relationship between volume and capacity is crucial: $1 \text{ cm}^3 = 1 \text{ ml}$ and $1000 \text{ cm}^3 = 1 \text{ litre}$. Visually, imagine a $10 \text{ cm} \times 10 \text{ cm} \times 10 \text{ cm}$ cube filling exactly a 1-litre bottle.

Unit conversion is essential in volume calculations. To convert $m^3$ to $cm^3$, multiply by $10,00,000$ (since $1 \text{ m} = 100 \text{ cm}$, then $1 \text{ m}^3 = 100^3 \text{ cm}^3$). Always ensure all dimensions are in the same units before calculating.