Surface Area of a Cylinder

Statement

The surface area of a cylinder is the total area of its curved surface plus the area of its two circular ends. A cylinder has two key measurements: the radius \(r\) of the circular base and the height \(h\) of the curved side. The formula for the total surface area is:

\[ S = 2\pi r h + 2\pi r^2 \]

The first term \(2\pi r h\) gives the curved surface area (the “label” wrapped around the cylinder), while the second term \(2\pi r^2\) gives the combined area of the two circular ends (top and bottom).

Why it’s true

• The cylinder’s side can be “unwrapped” into a rectangle with width equal to the circumference of the circle (\(2\pi r\)) and height equal to the cylinder’s height (\(h\)). So its area is \(2\pi r h\).
• Each end is a circle of radius \(r\), area \(\pi r^2\). There are two ends, so their total area is \(2\pi r^2\).
• Adding these together gives the formula: \(S = 2\pi r h + 2\pi r^2\).

Recipe (how to use it)

1. Identify the radius \(r\) and the height \(h\).
2. Calculate the curved surface area: \(2\pi r h\).
3. Calculate the area of the two circles: \(2\pi r^2\).
4. Add them together for the total surface area.
5. Give the final answer in square units (cm², m², etc.).

Spotting it

Use this formula whenever the problem involves a cylinder and asks for its total surface area (e.g., tins, pipes, tubes). If only the curved part is needed (like labelling a can), use \(2\pi r h\) alone.

Common pairings

• Volume of a cylinder: \(V = \pi r^2 h\).
• Area of a circle: \(A = \pi r^2\).

Mini examples

1. Given: \(r=3\), \(h=10\). Find: \(S\). Answer: \(2\pi \times 3 \times 10 + 2\pi \times 9 = 60\pi + 18\pi = 78\pi \approx 245.0\).
2. Given: \(r=5\), \(h=7\). Find: \(S\). Answer: \(2\pi \times 5 \times 7 + 2\pi \times 25 = 70\pi + 50\pi = 120\pi \approx 376.99\).

Pitfalls

• Forgetting to include the circular ends when asked for total surface area.
• Mixing up diameter and radius.
• Leaving the answer without square units.

Exam strategy

• Write each step separately to avoid dropping a term.
• If an exact form is required, leave your answer in terms of \(\pi\).
• If a decimal is required, use a calculator and round to 1 or 2 decimal places depending on the question.

Summary

The surface area of a cylinder includes both the curved surface and the circular ends. Always check whether the question wants the total surface area or just the curved part. Use the formula \(S = 2\pi r h + 2\pi r^2\), keep track of units, and decide whether to give your answer in exact terms or as a rounded decimal.