Surface Area of a Cylinder: Formula, Worked Examples and Practice Questions
The surface area of a cylinder question appears in almost every geometry paper from GCSE through to A Level foundation, and the marks lost on it are almost always for the same reasons: using the wrong formula, confusing curved surface area with total surface area or rounding too early.
This guide covers both formulas, derives them so you understand where they come from and walks through worked examples at increasing difficulty.
Parts of a cylinder
A cylinder has three surfaces:
• Two circular faces: one at the top, called the top face, and one at the bottom, called the base. Both are identical circles with radius r.
• One curved surface: the tube shape that wraps around the outside between the two circles. Its area is what you would get if you unrolled it flat into a rectangle.
The height of the cylinder is labelled h. The radius of each circular face is labelled r.
The two formulas you need
Surface | Formula | What it includes |
Curved surface area (CSA) | 2 pi r h | The tube only, no circles |
Total surface area (TSA) | 2 pi r h + 2 pi r squared | The tube plus both circular ends |
Total surface area simplified | 2 pi r (h + r) | The same formula factorised |
Choosing between CSA and TSA depends on what the cylinder represents. A tin can open at both ends needs only the curved surface area. A tin can with a lid and a base needs the total surface area. A swimming pool modelled as a cylinder open at the top needs the curved surface area plus one circular base. Always read the question carefully.
Where the formulas come from
Curved surface area
Imagine cutting the curved surface along a vertical line and unrolling it flat. You get a rectangle. Its width is the height of the cylinder, h. Its length is the circumference of the circular end, which is 2 pi r. Area of a rectangle equals length multiplied by width, so:
Curved surface area equals 2 pi r multiplied by h, which gives 2 pi r h.
Total surface area
Add the two circular faces to the curved surface. Each circle has area pi r squared, so two circles give 2 pi r squared. Adding the curved surface:
Total surface area equals 2 pi r h plus 2 pi r squared, which factors to 2 pi r (h plus r).
Understanding the derivation matters because it helps you reconstruct the formula if you forget it under exam pressure. A formula you can build is far more reliable than one you can only recall.
Worked example 1: curved surface area
Find the curved surface area of a cylinder with radius 4 cm and height 9 cm. Give your answer to one decimal place.
Using CSA equals 2 pi r h:
1. Substitute values: CSA equals 2 multiplied by pi multiplied by 4 multiplied by 9
2. Calculate: 2 multiplied by 4 multiplied by 9 equals 72
3. Multiply by pi: 72 multiplied by pi equals 226.194...
4. Round: 226.2 cm squared
Worked example 2: total surface area
A cylindrical can has a radius of 3.5 cm and a height of 12 cm. Find its total surface area, giving your answer to 3 significant figures.
Using TSA equals 2 pi r h plus 2 pi r squared:
5. Curved surface area: 2 multiplied by pi multiplied by 3.5 multiplied by 12 equals 84 pi
6. Two circles: 2 multiplied by pi multiplied by 3.5 squared equals 2 multiplied by pi multiplied by 12.25 equals 24.5 pi
7. Total: 84 pi plus 24.5 pi equals 108.5 pi
8. 108.5 multiplied by pi equals 340.869...
9. Rounded to 3 significant figures: 341 cm squared
Keeping terms in pi for as long as possible avoids rounding errors accumulating through the calculation. Only convert to a decimal in the final step.
Worked example 3: finding the radius from the surface area
The total surface area of a cylinder is 150 pi cm squared. Its height is 8 cm. Find its radius.
Using TSA equals 2 pi r (h plus r) and substituting:
10. 150 pi equals 2 pi r (8 plus r)
11. Divide both sides by 2 pi: 75 equals r (8 plus r)
12. Expand: 75 equals 8r plus r squared
13. Rearrange: r squared plus 8r minus 75 equals 0
14. Factorise: (r plus 15)(r minus 5) equals 0
15. Solutions: r equals negative 15 or r equals 5
16. Since radius cannot be negative, r equals 5 cm
Worked example 4: hollow or open cylinders
A cylindrical tube is open at both ends. Its outer radius is 5 cm, its height is 20 cm and its wall thickness is 1 cm. Find the total surface area.
Inner radius equals 5 minus 1 equals 4 cm.
17. Outer curved surface: 2 pi multiplied by 5 multiplied by 20 equals 200 pi
18. Inner curved surface: 2 pi multiplied by 4 multiplied by 20 equals 160 pi
19. Two annular rings (top and bottom): each ring has area pi multiplied by (5 squared minus 4 squared) equals pi multiplied by (25 minus 16) equals 9 pi. Two rings give 18 pi.
20. Total: 200 pi plus 160 pi plus 18 pi equals 378 pi
21. 378 multiplied by pi equals 1187.5 cm squared (to 1 decimal place)
Common mistakes to avoid
• Using diameter instead of radius. If the question gives diameter, halve it before substituting into the formula.
• Using the curved surface area formula when the total is needed, or vice versa. Read the question twice.
• Squaring only the r in 2 pi r squared and forgetting that the 2 is a coefficient, not part of the power.
• Rounding intermediate values, which builds errors into every subsequent step.
• Giving units as cm rather than cm squared. Surface area is always a square unit.
• In a composite shape, forgetting to subtract the circular face where two surfaces join.
Units and conversions
Always state your unit. Surface area is expressed in square units: mm squared, cm squared, m squared and so on. If the radius and height are given in different units, convert before substituting. Mixing centimetres and metres in the same calculation is one of the most reliable ways to lose a correct method mark.
Practice questions
22. Find the curved surface area of a cylinder with radius 7 cm and height 10 cm. Give your answer to 1 decimal place.
23. Find the total surface area of a cylinder with diameter 6 cm and height 14 cm. Give your answer to 3 significant figures.
24. The curved surface area of a cylinder is 188.5 cm squared and its height is 5 cm. Find the radius, correct to 1 decimal place.
25. A solid cylinder has radius 3 cm and height 8 cm. It is melted and recast into a cylinder with radius 2 cm. Find the total surface area of the new cylinder to 3 significant figures.
Answers
26. 2 multiplied by pi multiplied by 7 multiplied by 10 equals 439.8 cm squared
27. Radius equals 3 cm. TSA equals 2 pi multiplied by 3 multiplied by 14 plus 2 pi multiplied by 9 equals 84 pi plus 18 pi equals 102 pi equals 320 cm squared (to 3 s.f.)
28. 188.5 equals 2 pi r multiplied by 5, so r equals 188.5 divided by (10 pi) equals 6.0 cm
29. Volume of original equals pi multiplied by 9 multiplied by 8 equals 72 pi. New cylinder: pi multiplied by 4 multiplied by h equals 72 pi, so h equals 18 cm. TSA equals 2 pi multiplied by 2 multiplied by 18 plus 2 pi multiplied by 4 equals 72 pi plus 8 pi equals 80 pi equals 251 cm squared
Frequently Asked Questions
Q. What is the difference between curved surface area and total surface area of a cylinder?
A. Curved surface area covers only the tube shaped side of the cylinder, with no circular faces. Total surface area includes the curved surface plus both circular ends. Use curved when the circles are open or absent, and total when the cylinder is completely enclosed.
Q. What is the formula for the surface area of a cylinder?
A. Curved surface area equals 2 pi r h, where r is the radius and h is the height. Total surface area equals 2 pi r h plus 2 pi r squared, which simplifies to 2 pi r multiplied by (h plus r).
Q. How do I find the surface area of a cylinder if I am given the diameter?
A. Divide the diameter by 2 to get the radius, then use that radius in the formula. The formula always uses the radius, not the diameter.
Q. How do I find the radius if the surface area is given?
A. Substitute the known values into the appropriate formula and solve the resulting equation. If only the curved surface area is known, the equation is linear in r. If the total surface area is known, you will get a quadratic equation to solve.
Q. Should I use 3.14 or the pi button for surface area calculations?
A. Use the pi button on a calculator for accuracy throughout the calculation and only round in the final step, to the precision the question specifies. Using 3.14 or 22 over 7 introduces rounding error that accumulates.
Disclaimer
This article is for general educational guidance. Always follow the rounding rules and notation required by your own exam board.
