Mensuration - Volume of Cube, Cuboid, and Cylinder

Volume is the measure of the amount of three-dimensional space occupied by an object. To visualize this, imagine the total amount of water required to completely fill a hollow 3D container; that quantity represents its volume.

The standard units for measuring volume are cubic units, such as cubic centimeters ($cm^3$) or cubic meters ($m^3$). This is because volume is derived from the product of three linear dimensions (length, width, and height).

A cuboid is a solid bounded by six rectangular faces. To visualize its volume, consider a rectangular box with length $l$, breadth $b$, and height $h$; the total space inside the box is the product of these three dimensions.

A cube is a specific type of cuboid where all edges are of equal length, denoted as $a$. Visually, it looks like a standard die where the length, breadth, and height are identical, resulting in a volume of $a \times a \times a$.

A cylinder consists of two congruent circular bases connected by a curved surface. To visualize its volume, imagine a stack of identical circular coins. The total volume is the area of the bottom circular base multiplied by the total height $h$ of the stack.

The general principle for finding the volume of any uniform solid (like a prism or cylinder) is multiplying the Area of the Base by the perpendicular Height. This is expressed as $V = \text{Area of Base} \times h$.

Capacity refers to the volume of a substance (like liquid) that a container can hold. While volume measures the space an object occupies, capacity specifically measures the interior space of hollow objects, often measured in liters ($L$) or milliliters ($ml$).

Units of capacity are related to cubic units: $1000 cm^3 = 1 \text{ liter}$ and $1 m^3 = 1000 \text{ liters}$. Another useful conversion is $1 cm^3 = 1 ml$.