Surface Areas and Volumes - Volume of a Cylinder

Definition of a Cylinder: A right circular cylinder is a three-dimensional solid with two identical, parallel circular bases connected by a curved surface. Visually, it can be imagined as a stack of thin, circular coins of the same size placed one above the other to a certain height.

Radius and Height: The dimensions of a cylinder are defined by the radius ($r$) of its circular base and its height ($h$), which is the perpendicular distance between the centers of the two bases. If you look at the cylinder from the top, you see a circle; from the side, it appears as a rectangle with width equal to the diameter ($2r$) and height ($h$).

Concept of Volume: Volume represents the amount of space occupied by the cylinder. For any uniform solid like a cylinder, the volume is calculated by multiplying the area of the base by the height. This represents 'filling' the area of the circle throughout the entire vertical extent of the cylinder.

Base Area Relationship: Since the base of the cylinder is a circle, its area is $\pi r^2$. Therefore, the volume formula is essentially Area of Base $\times$ Height, which is $\pi r^2 \times h$.

Capacity: The term capacity is often used when dealing with hollow cylinders like pipes, tanks, or glasses. It refers to the internal volume or the amount of liquid/gas the cylinder can hold. For a hollow cylinder with negligible thickness, the volume and capacity are numerically the same.

Units of Measurement: Since volume is a three-dimensional measure, it is expressed in cubic units such as $cm^3$, $m^3$, or $mm^3$. It is important to ensure both radius and height are in the same units before calculation.

Unit Conversions for Liquids: In many practical problems, volume needs to be converted to liquid capacity. Common conversions include $1,000 cm^3 = 1 liter$, $1 m^3 = 1,000 liters$, and $1 cm^3 = 1 milliliter$.