1. Cylinder
A right circular cylinder is a 3D solid with:
- Two circular bases of radius \(r\)
- A height \(h\) joining the two bases
The curved surface can be imagined as the label wrapped around a cold drink can. When unrolled, this curved part becomes a rectangle.
This makes surface area calculations very simple and visual.
2. Curved (Lateral) Surface Area of a Cylinder
The curved surface area (CSA) is the area of the side curved surface only, without including the top and bottom circles.
If we cut the curved surface vertically and open it out, we get a rectangle with:
- Height = \(h\)
- Width = circumference of base = \(2\pi r\)
So the area of this rectangle is:
\( \text{CSA} = 2\pi r h \)
2.1. Example
A cylinder has radius \(r = 3\,\text{cm}\) and height \(h = 10\,\text{cm}\).
Curved surface area:
\( \text{CSA} = 2\pi r h = 2 \times \pi \times 3 \times 10 = 60\pi \)
Using \(\pi = 3.14\):
\( 60 \times 3.14 = 188.4\,\text{cm}^2 \)
3. Total Surface Area of a Cylinder
The Total Surface Area (TSA) includes both circular bases plus the curved surface.
The two bases have area:
\( 2 \times \pi r^2 \)
Adding this to the curved surface area gives:
\( \text{TSA} = 2\pi r h + 2\pi r^2 \)
You can factor this as:
\( \text{TSA} = 2\pi r (h + r) \)
3.1. Example
For a cylinder with \(r = 4\,\text{cm}\) and \(h = 7\,\text{cm}\):
\( \text{TSA} = 2\pi r (h + r) = 2\pi \times 4 \times (7 + 4) = 8\pi \times 11 = 88\pi \)
Using \(\pi = 3.14\):
\( 88 \times 3.14 = 276.32\,\text{cm}^2 \)
4. Where Cylinder Surface Areas Are Used
Cylinders appear often in practical problems. Some real-life uses:
- CSA → to find material for labels wrapped around bottles or cans
- TSA → to paint a cylindrical tank from outside
- TSA → to find the total cloth needed for a cylindrical pillar cover
- CSA → to calculate heat loss through the walls of a cylinder
Choosing CSA or TSA depends on whether the bases are included in the surface calculation.