MathsICSE Class 10 · Area and Volume of Solids – Cylinder, Cone and Sphere

Surface Area and Volume of a Right Circular Cylinder, Including Hollow and Cut Cylinders (ICSE Class 10)

Q: What is the difference between CSA and TSA of a cylinder?

The curved surface area 2πrh is only the side. The total surface area 2πr(r + h) adds the two circular ends.

Q: How do I find the surface area of a cylinder cut into two halves?

Take the total surface area of the whole cylinder and add the two new rectangular faces, each of size diameter × height.

Q: What value of π should I use in ICSE?

Use the value the question gives, usually 22/7. If none is given, 22/7 or 3.14 is acceptable; round only at the end.

Updated 15 October 2026 · Fact-checked

For a cylinder of radius $r$ and height $h$: curved surface area $= 2\pi rh$, total surface area $= 2\pi r(r + h)$ and volume $= \pi r^2 h$. A hollow cylinder with radii $R$ and $r$ has volume $\pi(R^2 - r^2)h$. Cutting a solid creates new flat faces that must be added.

Unrolled, the curved surface is a rectangle of 2πr by h, so CSA = 2πrh; add two circles of πr² each for TSA.

Illustration drawn for QuestionBuddy.

Understand Surface Area and Volume of a Cylinder

The CISCE syllabus covers surface area and volume of solids: cylinder, cone and sphere (frustum excluded). This page is the cylinder.

The net of a cylinder. Unroll the curved surface and you get a rectangle: one side is the height $h$ , the other is the circumference $2 π r$ . So

Curved surface area (CSA) $= 2 π r h$ ,
Total surface area (TSA) $= 2 π r h + 2 π r^{2} = 2 π r (r + h)$ (closed at both ends),
Volume $= π r^{2} h$ (area of the base × height).

An open cylinder (a tank without a lid, a pipe) uses only the faces that are actually there: open at one end $= 2 π r h + π r^{2}$ .

Hollow cylinder (pipe). External radius $R$ , internal radius $r$ , thickness $R - r$ :

Volume of material $= π (R^{2} - r^{2}) h$ .
Total surface area $= 2 π R h + 2 π r h + 2 π (R^{2} - r^{2})$ (outside, inside and two rings).

Cutting a cylinder. Cutting along the height into two halves adds two rectangles, each $2 r \times h$ . In 2024 Q2(ii) (diameter 7 cm, height 10 cm) the total for both halves was $297 + 140 = 437$ cm²; the examiners report many candidates left out the new faces.

Rolling a sheet. A rectangle $a \times b$ rolled so that side $a$ becomes the circumference gives $2 π r = a$ and $h = b$ . The CSA is always the area of the sheet, $ab$ , whichever way it is rolled, but the volumes differ. This was 2024 Q1(xiv).

Units and conversions. $1 m^{3} = 1000$ litres, $1000 cm^{3} = 1$ litre, $1 m^{2} = 10000 cm^{2}$ . A roller covers its CSA in one revolution.

Ratios. For two cylinders, volumes are in the ratio $r_{1}^{2} h_{1} : r_{2}^{2} h_{2}$ and CSAs in the ratio $r_{1} h_{1} : r_{2} h_{2}$ .

Surface Area and Volume of a Cylinder: formulas to remember

Curved surface area: $CSA = 2 π r h$; Area of the unrolled rectangle.
Total surface area: $TSA = 2 π r (r + h)$; Closed at both ends.
Volume: $V = π r^{2} h$; Base area × height.
Hollow cylinder: $V = π (R^{2} - r^{2}) h$; $R$ external, $r$ internal radius.
Two halves cut along the height: $TSA_{both} = 2 π r (r + h) + 2 (2 r \times h)$; Two new rectangular faces.

How to solve Surface Area and Volume of a Cylinder questions

Picture the solid, list every face that is really there, then choose the formula.

1Write down $r$ and $h$ ; halve any diameter first.
2Decide what is asked: CSA (label, roller, inside of a pipe), TSA (closed solid) or volume (capacity, material).
3For hollow or cut solids, list each surface: outer curved, inner curved, rings, new flat faces.
4Substitute with $π = \frac{22}{7}$ unless told otherwise, and simplify before multiplying.
5Convert units at the end (cm³ to litres, cm² to m²) and give the cost or mass if asked.

Surface Area and Volume of a Cylinder in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Capacity of a bottle

Suppose a cylindrical water bottle has an internal radius of 3.5 cm and a height of 20 cm.

How many litres does it hold? ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

$V = \frac{22}{7} \times 3.5 \times 3.5 \times 20 = 770$ cm³.
$770 \div 1000 = 0.77$ litres.

Answer: 0.77 litres

Memory hook: Volume = base circle × height.

Curved surface area of a label

Imagine a paper label wrapped all the way round a tin of radius 3.5 cm, covering a height of 12 cm.

How much paper does the label need, ignoring overlap? ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

The label covers only the curved surface, not the ends, so use $2 π r h$ .
$2 \times \frac{22}{7} \times 3.5 \times 12 = 22 \times 12 = 264$ cm².

Answer: 264 cm²

Memory hook: Label = unrolled rectangle = circumference × height.

Tricks to remember Surface Area and Volume of a Cylinder

Short memory aids. Say each one aloud once, then use it on a question the same day.

Unroll it
The curved surface is a rectangle, $2 π r$ by $h$ .
Use it when: Rolled-sheet and label questions.
Count the faces
Cut it? New faces. Hollow? Inside, outside and two rings. Open? Fewer circles.
Use it when: TSA of non-standard cylinders.
Squared radius, single height
Volume scales with $r^{2}$ but only with $h$ .
Use it when: Ratio and 'what if the radius doubles' questions.

Surface Area and Volume of a Cylinder: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Volume from radius and height · 1 mark · MCQBoard level

The volume of a cylinder of radius 3.5 cm and height 10 cm is ( $π = \frac{22}{7}$ ):
(a) 110 cm³
(b) 220 cm³
(c) 385 cm³
(d) 770 cm³

Try it first, then show the solutionSolution

$V = \frac{22}{7} \times 3.5 \times 3.5 \times 10 = 11 \times 3.5 \times 10 = 385$ cm³.

Answer: (c)

Example 2: CSA, TSA and volume together · 3 marks · Short answerBoard level

A closed cylinder has diameter 21 cm and height 15 cm. Find its curved surface area, total surface area and volume. ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

$r = 10.5$ cm. CSA $= 2 \times \frac{22}{7} \times 10.5 \times 15 = 66 \times 15 = 990$ cm².
Each end $= \frac{22}{7} \times 10. 5^{2} = 346.5$ cm², so TSA $= 990 + 693 = 1683$ cm².
$V = 346.5 \times 15 = 5197.5$ cm³.

Answer: CSA 990 cm², TSA 1683 cm², volume 5197.5 cm³

Example 3: A hollow cylinder · 4 marks · Long answerHarder board level

A hollow metal cylinder is 14 cm long, with external radius 6 cm and internal radius 4 cm. Find (a) the volume of metal, (b) its total surface area (outside, inside and both ends), to 2 decimal places. ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

(a) $V = \frac{22}{7} \times (6^{2} - 4^{2}) \times 14 = \frac{22}{7} \times 20 \times 14 = 880$ cm³.
(b) Outer CSA $= 2 \times \frac{22}{7} \times 6 \times 14 = 528$ cm²; inner CSA $= 2 \times \frac{22}{7} \times 4 \times 14 = 352$ cm².
Two rings $= 2 \times \frac{22}{7} \times (36 - 16) = \frac{880}{7} \approx 125.71$ cm².
Total $= 528 + 352 + 125.71 = 1005.71$ cm².

Answer: (a) 880 cm³ (b) 1005.71 cm²

Example 4: Cut into two halves · 4 marks · Long answerHarder board level

A solid cylinder of diameter 14 cm and height 9 cm is cut into two identical halves along its height. Find the total surface area of the two halves and the cost of painting them at ₹5 per cm². ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

Original TSA $= 2 \times \frac{22}{7} \times 7 \times (7 + 9) = 704$ cm².
The cut makes two new rectangular faces, each $14 \times 9 = 126$ cm².
Total $= 704 + 2 \times 126 = 956$ cm².
Cost $= 956 \times 5$ = ₹4780.

Answer: 956 cm²; ₹4780

Example 5: One sheet, two cylinders · 4 marks · Long answerHOTS / competency

A rectangular sheet 44 cm × 22 cm is rolled into a hollow cylinder (a) along its length, so that the 44 cm side becomes the circumference, and (b) along its width, so that the 22 cm side becomes the circumference. Compare the curved surface areas and the volumes. ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

(a) $2 π r = 44$ gives $r = 7$ cm, $h = 22$ cm: $V = \frac{22}{7} \times 49 \times 22 = 3388$ cm³.
(b) $2 π r = 22$ gives $r = 3.5$ cm, $h = 44$ cm: $V = \frac{22}{7} \times 12.25 \times 44 = 1694$ cm³.
Both CSAs equal the sheet's area, $44 \times 22 = 968$ cm²: ratio $1 : 1$ .
Volumes: $3388 : 1694 = 2 : 1$ . The wider cylinder holds twice as much.

Answer: CSAs equal (968 cm²); volumes 3388 cm³ and 1694 cm³, ratio $2 : 1$

Example 6: Case study: rainwater harvesting · 4 marks · Case-basedHOTS / competency

Rain falls on a flat roof measuring 22 m × 20 m. All the water from a 2 cm rainfall is collected in a cylindrical tank of radius 1 m.
(a) Find the volume of water collected, in m³ and in litres.
(b) How high does the water rise in the tank?
(c) The tank is 3 m tall. Will it overflow? ( $π = \frac{22}{7}$ )

Try it first, then show the solutionSolution

(a) $V = 22 \times 20 \times 0.02 = 8.8$ m³ $= 8800$ litres.
(b) $\frac{22}{7} \times 1^{2} \times h = 8.8$ , so $h = \frac{8.8 \times 7}{22} = 2.8$ m.
(c) $2.8 < 3$ , so it does not overflow (0.2 m to spare).

Answer: (a) 8.8 m³ = 8800 L (b) 2.8 m (c) No

Where students lose marks in Surface Area and Volume of a Cylinder

The slips examiners see most often, and what to write instead.

Mistake: Leaving out the new faces after cutting
A cut along the height exposes two rectangles of size $2 r \times h$ .
Wrong working
TSA of both halves $= 2 π r (r + h) = 297$ cm² only.
Right working
$297 + 2 \times (7 \times 10) = 437$ cm².
Do this: Sketch the pieces and list every face before calculating.
Usually costs: Reported in 2024 Q2(ii): candidates got 297 cm² or found one half only.
Mistake: Diameter used as radius
Formulas use the radius.
Wrong working
Diameter 7 cm, so $r = 7$ cm.
Right working
$r = 3.5$ cm.
Do this: Write ' $r = \dots$ ' as the first line.
Usually costs: Reported in 2024 Q2(ii): radius taken as 7 cm.
Mistake: Swapping radius and height when a sheet is rolled
The side that wraps round becomes $2 π r$ ; the other side is $h$ .
Wrong working
Sheet 44 × 22 rolled along the length: $r = 22$ , $h = 44$ .
Right working
$2 π r = 44$ , $r = 7$ , $h = 22$ .
Do this: Ask 'which edge goes round?' before writing $r$ .
Usually costs: Reported in 2024 Q1(xiv): candidates confused the formula when radius and height interchange.

Exam tips

2024 Q2(ii) (4 marks): two halves of a cylinder, $d = 7$ , $h = 10$ : $297 + 140 = 437$ cm², cost $437 \times 30$ = ₹13 110.
2024 Q1(xiv) (1 mark): an 11 × 7 sheet rolled two ways gives equal curved surface areas, ratio $1 : 1$ .
Use $π = \frac{22}{7}$ when the radius is a multiple of 7 or 3.5; cancel before multiplying.
Give units: cm² for areas, cm³ for volumes, and convert to litres or m² only when asked.

Practice questions on Surface Area and Volume of a Cylinder

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The radii of two cylinders are in the ratio $2 : 3$ and their heights in the ratio $5 : 4$ . The ratio of their volumes is:
(a) $5 : 6$
(b) $5 : 9$
(c) $10 : 9$
(d) $20 : 27$
Hint
$r_{1}^{2} h_{1} : r_{2}^{2} h_{2}$ .
Answer
(b): $4 \times 5 : 9 \times 4 = 20 : 36$ .
Q2 · 3 marks · Short answerBoard level
The curved surface area of a cylinder is 1320 cm² and its radius is 10.5 cm. Find its height and volume. ( $π = \frac{22}{7}$ )
Hint
$2 π r = 66$ .
Answer
$66 h = 1320$ , $h = 20$ cm; $V = 346.5 \times 20 = 6930$ cm³.
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If the radius of a cylinder is halved and its height is doubled, its curved surface area does not change.
Reason (R): The curved surface area of a cylinder is $2 π r h$ .
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$2 π \cdot \frac{r}{2} \cdot 2 h$ .
Answer
(c)
Q4 · 3 marks · Short answerHarder board level
A closed cylinder of height 9 cm has total surface area 704 cm². Find its radius and volume. ( $π = \frac{22}{7}$ )
Hint
$r (r + 9) = 112$ .
Answer
$2 \times \frac{22}{7} r (r + 9) = 704$ gives $r^{2} + 9 r - 112 = 0$ , $r = 7$ cm; $V = 154 \times 9 = 1386$ cm³.
Q5 · 3 marks · Short answerHarder board level
An iron rod is 2 m long and 1.4 cm in diameter. Find its volume and its mass, if 1 cm³ of iron weighs 7.8 g. ( $π = \frac{22}{7}$ )
Hint
Use $h = 200$ cm.
Answer
$V = \frac{22}{7} \times 0.49 \times 200 = 308$ cm³; mass $= 2402.4$ g $\approx 2.40$ kg.
Q6 · 3 marks · Short answerHarder board level
A garden roller has diameter 84 cm and length 120 cm. Find the area it levels in 500 complete revolutions, in m², and the cost of levelling at ₹2 per m². ( $π = \frac{22}{7}$ )
Hint
One revolution covers the CSA.
Answer
CSA $= 2 \times \frac{22}{7} \times 42 \times 120 = 31680$ cm²; 500 revolutions $= 15840000$ cm² $= 1584$ m²; cost ₹3168.
Q7 · 4 marks · Long answerHOTS / competency
A hollow cylindrical pipe is 14 cm long with external radius 5 cm. The volume of metal in it is 704 cm³. Find its internal radius and its thickness. ( $π = \frac{22}{7}$ )
Hint
$π (R^{2} - r^{2}) h = 704$ .
Answer
$44 (25 - r^{2}) = 704$ , so $25 - r^{2} = 16$ and $r = 3$ cm; thickness $= 2$ cm.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.CSA of a cylinder?Show answer
$2 π r h$
2.TSA of a closed cylinder?Show answer
$2 π r (r + h)$
3.Volume of a hollow cylinder?Show answer
$π (R^{2} - r^{2}) h$
4. $r = 7$ cm, $h = 10$ cm. Volume?Show answer
1540 cm³
5.How many litres in 1 m³?Show answer
1000

Surface Area and Volume of a Cylinder: frequently asked questions

What is the difference between CSA and TSA of a cylinder?

The curved surface area 2πrh is only the side. The total surface area 2πr(r + h) adds the two circular ends.

How do I find the surface area of a cylinder cut into two halves?

Take the total surface area of the whole cylinder and add the two new rectangular faces, each of size diameter × height.

What value of π should I use in ICSE?

Use the value the question gives, usually 22/7. If none is given, 22/7 or 3.14 is acceptable; round only at the end.