Mensuration - Volume and Capacity

Volume is the measure of the three-dimensional space occupied by a solid object. Visually, you can imagine filling a 3D shape with unit cubes (cubes with side $1 unit$); the total number of these cubes that fit inside the object defines its volume.

Capacity refers to the quantity of liquid or substance that a hollow container can hold. While volume describes the space the object takes up, capacity describes the interior space available for contents. Visually, think of the difference between the thickness of a glass bottle (volume) and the amount of milk it can hold (capacity).

The volume of a Cuboid is determined by the product of its length, breadth, and height. Visually, it is like a rectangular base of area $l \times b$ that has been stretched upwards to a height $h$, creating a stack of rectangular layers.

The volume of a Cube is a special case of a cuboid where all sides are equal. Visually, it is a perfectly symmetrical shape like a die, where the length, breadth, and height are all the same value $a$, making the volume $a \times a \times a$.

The volume of a Cylinder is calculated by multiplying the area of its circular base by its height. Visually, imagine a stack of identical circular coins; the area of one coin is $\pi r^2$, and the height of the stack is $h$.

A fundamental principle for uniform solids (like prisms and cylinders) is that Volume = Area of Base $\times$ Height. This visual concept helps in finding the volume of any shape that has the same cross-section from bottom to top.

Units of measurement for volume are cubic units such as $mm^3$, $cm^3$, or $m^3$. For capacity, units like milliliters ($mL$) and liters ($L$) are used. Visually, $1 cm^3$ is exactly the same size as $1 mL$.

Conversion between volume and capacity is essential: $1000 cm^3 = 1 L$ and $1 m^3 = 1000 L$. Visually, a large tank of $1 meter$ length, width, and height would hold exactly $1000$ liters of water.