The total surface area of a closed cylinder is the combined area of its two circular ends plus its curved rectangular side: SA = 2πr² + 2πrh, where r is the radius of the circular cross-section and h is the height (length) of the cylinder. The formula combines two familiar area calculations into a single expression.
Why does the cylinder surface area formula have two parts?
Imagine "unrolling" a closed cylinder. You get:
- Two circles (the top and bottom faces), each with area πr².
- One rectangle (the curved side laid flat), with width equal to the circumference of the circle (2πr) and length equal to the height h.
So the total surface area is:
$$SA = \underbrace{2 \times \pi r^2}{\text{two circular ends}} + \underbrace{2\pi r h}{\text{curved side}}$$
This can be factorised: SA = 2πr(r + h).
How do you calculate the total surface area of a closed cylinder?
Step-by-step method:
- Identify the radius r and height h. (If given the diameter, halve it to find r.)
- Calculate the area of one circular end:
πr². - Calculate the curved surface area:
2πrh. - Add: Total SA = 2πr² + 2πrh.
- State your units (always area units — cm², m², etc.).
Worked example 1: closed cylinder
Find the total surface area of a closed cylinder with radius 4 cm and height 9 cm. Give your answer in terms of π.
- Area of one circle:
π × 4² = 16π cm² - Two circular ends:
2 × 16π = 32π cm² - Curved surface area:
2π × 4 × 9 = 72π cm² - Total SA =
32π + 72π = **104π cm²** ≈ 326.7 cm²
Worked example 2: cylinder with given diameter
A tin can has a diameter of 7 cm and a height of 12 cm. Find the total surface area to 1 decimal place.
- Radius:
r = 7 ÷ 2 = 3.5 cm - Two circular ends:
2 × π × 3.5² = 2 × π × 12.25 = 24.5π ≈ 76.97 cm² - Curved surface area:
2 × π × 3.5 × 12 = 84π ≈ 263.89 cm² - Total SA ≈ `76.97 + 263.89 = 340.9 cm² (to 1 d.p.)
What is the curved surface area only?
For an open cylinder (no circular ends — like a tube or pipe), only the curved surface area is needed:
$$\text{Curved SA} = 2\pi r h$$
Example: A drainpipe has a radius of 5 cm and a length of 250 cm. Find the external surface area.
Curved SA = 2 × π × 5 × 250 = 2500π ≈ 7854 cm²
The two ends are not included because the pipe is open at both ends.
For a cylinder open at one end only (like a cup):
$$SA = \pi r^2 + 2\pi r h$$
(One circular base + curved side, no lid.)
Summary of the three cases
| Cylinder type | Formula | When to use |
|---|---|---|
| Closed (top and bottom) | 2πr² + 2πrh | Tin can, storage cylinder |
| Open at both ends | 2πrh | Pipe, tube, hollow barrel |
| Open at one end | πr² + 2πrh | Cup, beaker, bucket |
What mistakes should you avoid?
- Using the diameter instead of the radius. Always check: if you are given the diameter (the full width), divide by 2 to get r before substituting.
- Forgetting to double the circle area. The formula includes 2πr², not just πr² — there are two circular ends on a closed cylinder.
- Confusing surface area with volume. Volume = πr²h (area of cross-section × height). Surface area uses the circumference (2πr) in one of its parts.
- Omitting the units. Surface area is always expressed in square units: cm², m², mm².
Frequently asked questions
How do I find the surface area if given the volume?
If you know the volume (V = πr²h) and one dimension, find the missing dimension first, then apply the surface area formula. For example, if V = 150π cm³ and r = 5 cm, then h = 150π ÷ (π × 25) = 6 cm. Now SA = 2π(25) + 2π(5)(6) = 50π + 60π = 110π cm².
Is the curved surface area always larger than the two circles?
Not necessarily — it depends on the proportions. For a short, wide cylinder (large r, small h), the two circles dominate. For a long, narrow cylinder (small r, large h), the curved surface dominates. This is why drinks cans are designed to minimise total material (surface area) for a given volume — the optimal proportions are when h = 2r (height equals diameter).
Does the GCSE formula sheet include the cylinder surface area?
The AQA, OCR, and Edexcel formula sheets include the volume of a cylinder (V = πr²h) but not the surface area. You must remember SA = 2πr² + 2πrh, or derive it by recalling that the curved surface "unrolls" to a rectangle with dimensions 2πr × h.
Can I leave my answer as a multiple of π?
Yes — many GCSE questions say "give your answer in terms of π" or "leave your answer as a multiple of π". In that case, write 104π cm² rather than 326.7 cm². If no instruction is given, use a calculator and round to the required number of decimal places or significant figures.
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