1. Understanding a Cylinder
A right circular cylinder is a 3D shape with:
- a circular base of radius \(r\)
- a height \(h\) between the two bases
Cylinders occur often in real life—water tanks, pipes, cans, glasses, jars, batteries, and more. The volume tells us how much space or capacity the cylinder has.
2. Formula for Volume of a Cylinder
The cylinder can be thought of as a stack of several circular layers placed one above another. Each layer is a circle of area \(\pi r^2\).
If the height of the cylinder is \(h\), we are stacking h such layers.
So the volume becomes:
\( \text{Volume} = \pi r^2 h \)
2.1. Example
A cylinder has radius \(r = 4\,\text{cm}\) and height \(h = 10\,\text{cm}\). Then:
\( \text{Volume} = \pi r^2 h = \pi \times 4^2 \times 10 = 160\pi \)
Using \(\pi = 3.14\):
\( 160 \times 3.14 = 502.4\,\text{cm}^3 \)
This is the internal capacity of the cylinder.
3. Visual Understanding
Imagine slicing the cylinder horizontally into many thin circular discs.
- Each disc has area \(\pi r^2\)
- Height of the stack is \(h\)
So the total space is like stacking \(h\) circles on top of each other:
\( \text{Volume} = \pi r^2 h \)
4. Real-Life Applications
Volume of a cylinder is commonly used in:
- Finding the capacity of jars, bottles, and cylindrical tanks
- Calculating water in pipes
- Determining material needed to manufacture cylindrical rods
- Designing cans and packaging
Whenever the shape is round and tall like a pipe or container, use \(\pi r^2 h\) to measure its volume.