MathsCBSE Class 9 · Math of Space: Surface Area and Volume

Cuboids, Cubes and Cylinders: Surface Area and Volume, Class 9

Updated 17 October 2026 · Fact-checked

For a cuboid $l \times b \times h$: total surface area $= 2(lb + bh + hl)$, lateral surface area $= 2h(l+b)$, volume $= lbh$, diagonal $= \sqrt{l^2+b^2+h^2}$. For a cube of edge $a$: $6a^2$, $4a^2$, $a^3$, $a\sqrt{3}$. For a cylinder of radius $r$ and height $h$: curved surface $2\pi rh$, total surface $2\pi r(r+h)$, volume $\pi r^2 h$. Keep units consistent and use $\pi = \frac{22}{7}$ unless told otherwise.

The curved surface unrolls into a rectangle 2πr by h, so CSA = 2πrh; adding two circular bases gives TSA = 2πr(r + h).

Illustration drawn for QuestionBuddy.

Understand Cuboids, Cubes and Cylinders: Surface Area and Volume

Cuboid and cube. A cuboid has six rectangular faces in three pairs. Its total surface area is the sum of all six faces, $2 (l b + bh + h l)$ ; the lateral surface area counts the four walls only, $2 h (l + b)$ . The volume is $l bh$ . The diagonal (corner to opposite corner through the inside) has length $l^{2} + b^{2} + h^{2}$ , by applying Pythagoras twice. For a cuboid $12 \times 9 \times 8$ the diagonal is $144 + 81 + 64 = 289 = 17$ . A cube is a cuboid with $l = b = h = a$ : total surface area $6 a^{2}$ , lateral surface area $4 a^{2}$ , volume $a^{3}$ .

Cylinder. A right circular cylinder of radius $r$ and height $h$ has two circular faces and a curved surface. Cut the curved surface along a line and unroll it: it is a rectangle of length $2 π r$ and breadth $h$ . So the curved surface area (CSA) is $2 π r h$ , the total surface area (TSA) is $2 π r h + 2 π r^{2} = 2 π r (r + h)$ and the volume is $π r^{2} h$ . An *open* tank has only one circular base, so the area to be covered is $2 π r h + π r^{2}$ .

Rolling a sheet. A rectangular sheet $44$ cm $\times$ $30$ cm can be rolled into a cylinder in two ways. If the $44$ cm side becomes the circumference, $2 π r = 44$ , so $r = 7$ cm and $h = 30$ cm: volume $= \frac{22}{7} \times 49 \times 30 = 4620$ cm $^{3}$ . If the $30$ cm side is the circumference, $r = \frac{30 \times 7}{2 \times 22} = \frac{105}{22}$ cm and $h = 44$ cm: the volume works out to $3150$ cm $^{3}$ . The curved surface area is the same in both ( $1320$ cm $^{2}$ ), but the volume differs, which is a useful test of understanding: surface area and volume are not tied together.

Hollow cylinders. For a pipe with outer radius $R$ , inner radius $r$ and length $h$ , the volume of material is $π (R^{2} - r^{2}) h$ . For $R = 5$ , $r = 4$ , $h = 14$ : $\frac{22}{7} \times 9 \times 14 = 396$ cm $^{3}$ .

Units and capacity. Area is in square units and volume in cubic units. $1$ m $^{3}$ $= 1000$ litres and $1$ cm $^{3} = 1$ mL. A tank $2.2$ m $\times$ $1.4$ m $\times$ $1$ m holds $3.08$ m $^{3} = 3080$ litres. Convert all lengths to one unit before multiplying.

Rollers and wheels. One full turn of a roller covers an area equal to its CSA. The number of turns to cover a ground of area $A$ is $\frac{A}{2 π r h}$ .

Method. (1) List the given quantities with units. (2) Decide which surfaces count (open, closed, lateral). (3) Substitute in the formula, simplify by cancelling 7 with $\frac{22}{7}$ early, and (4) give the final answer with the right unit. Check by estimating: $π \approx 3$ gives a rough value.

Cuboids, Cubes and Cylinders: Surface Area and Volume: formulas to remember

Cuboid: $TSA = 2 (l b + bh + h l), LSA = 2 h (l + b), V = l bh$; Diagonal $= l^{2} + b^{2} + h^{2}$ .
Cube: $TSA = 6 a^{2}, LSA = 4 a^{2}, V = a^{3}$; Diagonal $= a 3$ .
Cylinder curved surface: $CSA = 2 π r h$; Unrolls to a rectangle $2 π r \times h$ .
Cylinder total surface: $TSA = 2 π r (r + h)$; Open at one end: $2 π r h + π r^{2}$ .
Cylinder volume: $V = π r^{2} h$; Base area times height.
Hollow cylinder: $V = π (R^{2} - r^{2}) h$; Volume of material.
Capacity: $1 m^{3} = 1000 L, 1 cm^{3} = 1 mL$; Convert before dividing by a flow rate.

How to solve Cuboids, Cubes and Cylinders: Surface Area and Volume questions

Decide which surfaces and which volume the question needs, then substitute with units.

1Write the given dimensions with their units and convert them to one unit.
2Decide the surface needed: total, lateral/curved, or open (one base).
3Write the formula, substitute, and cancel factors of 7 early when $π = \frac{22}{7}$ .
4For recasting problems equate volumes; for rolling problems equate the side to $2 π r$ or to $h$ .
5For costs or times, multiply area by rate or divide volume by flow rate.
6State the answer with the unit and check by rough estimation.

Cuboids, Cubes and Cylinders: Surface Area and Volume 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.

Volume of a cylinder

Suppose a cylindrical water bottle has an inner radius of 3.5 cm and is filled to a height of 20 cm.

How many millilitres of water does it hold?

Try it first, then show the solutionSolution

Volume $= π r^{2} h = \frac{22}{7} \times 3.5 \times 3.5 \times 20$ .
$\frac{22}{7} \times 3.5 = 11$ , so $V = 11 \times 3.5 \times 20 = 770$ cm $^{3}$ .
$1$ cm $^{3} = 1$ mL.

Answer: The bottle holds 770 mL.

Memory hook: Base area times height; a cubic centimetre is a millilitre.

Tricks to remember Cuboids, Cubes and Cylinders: Surface Area and Volume

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

Unroll the cylinder
The curved surface is a rectangle $2 π r$ long and $h$ high.
Use it when: Remembering CSA and rolled-sheet problems.
Open or closed?
Count surfaces explicitly: a tank open at the top loses one circle.
Use it when: Open tanks, cups, buckets.
Pythagoras twice
Face diagonal first, then space diagonal.
Use it when: Cuboid diagonal.

Cuboids, Cubes and Cylinders: Surface Area and Volume: solved examples

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

Example 1: Cuboid · 3 marks · Short answerBoard level

A cuboid is 12 cm long, 9 cm wide and 8 cm high. Find its total surface area, its volume and the length of its space diagonal.

Try it first, then show the solutionSolution

TSA $= 2 (12 \times 9 + 9 \times 8 + 8 \times 12) = 2 (108 + 72 + 96) = 552$ cm $^{2}$ . (1)
Volume $= 12 \times 9 \times 8 = 864$ cm $^{3}$ . (1)
Diagonal $= 1 2^{2} + 9^{2} + 8^{2} = 289 = 17$ cm. (1)

Answer: TSA $552$ cm $^{2}$ , volume $864$ cm $^{3}$ , diagonal $17$ cm.

Example 2: Metal drum · 3 marks · Short answerBoard level

A closed cylindrical drum has a diameter of 28 cm and a height of 45 cm. Find the area of its curved surface, its total surface area and its volume. Use $π = \frac{22}{7}$ .

Try it first, then show the solutionSolution

Radius $r = 14$ cm. CSA $= 2 π r h = 2 \times \frac{22}{7} \times 14 \times 45 = 88 \times 45 = 3960$ cm $^{2}$ . (1)
Two circular ends: $2 π r^{2} = 2 \times \frac{22}{7} \times 196 = 1232$ cm $^{2}$ , so TSA $= 3960 + 1232 = 5192$ cm $^{2}$ . (1)
Volume $= π r^{2} h = \frac{22}{7} \times 196 \times 45 = 616 \times 45 = 27720$ cm $^{3}$ . (1)

Answer: CSA $3960$ cm $^{2}$ , TSA $5192$ cm $^{2}$ , volume $27720$ cm $^{3}$ .

Example 3: Rolling a sheet two ways · 5 marks · Long answerHarder board level

A rectangular sheet of paper $44$ cm $\times$ $30$ cm is rolled into an open cylinder in two ways: with the 44 cm side as the circumference, or with the 30 cm side as the circumference. Find the volume in each case and the difference. Use $π = \frac{22}{7}$ .

Try it first, then show the solutionSolution

Case 1: $2 π r = 44$ gives $r = 7$ cm, height $30$ cm. (1)
$V_{1} = \frac{22}{7} \times 49 \times 30 = 4620$ cm $^{3}$ . (1)
Case 2: $2 π r = 30$ gives $r = \frac{105}{22}$ cm, height $44$ cm. (1)
$V_{2} = \frac{22}{7} \times (\frac{105}{22})^{2} \times 44 = \frac{11025}{154} \times 44 = 3150$ cm $^{3}$ . (1)
Difference $= 4620 - 3150 = 1470$ cm $^{3}$ ; the first cylinder (wider, shorter) holds more. (1)

Answer: $V_{1} = 4620$ cm $^{3}$ , $V_{2} = 3150$ cm $^{3}$ ; the difference is $1470$ cm $^{3}$ .

Example 4: Hollow pipe · 3 marks · Short answerHarder board level

A pipe has outer radius 5 cm, inner radius 4 cm and length 14 cm. Find the volume of material in it and the area of its inner curved surface. Use $π = \frac{22}{7}$ .

Try it first, then show the solutionSolution

Volume of material $= π (R^{2} - r^{2}) h = \frac{22}{7} (25 - 16) \times 14$ . (1)
$= \frac{22}{7} \times 9 \times 14 = 396$ cm $^{3}$ . (1)
Inner curved surface $= 2 π r h = 2 \times \frac{22}{7} \times 4 \times 14 = 352$ cm $^{2}$ . (1)

Answer: Volume $396$ cm $^{3}$ ; inner curved surface $352$ cm $^{2}$ .

Example 5: Same capacity, which uses less metal? · 4 marks · Case-basedHOTS / competency

Case: a rectangular closed tank measures $2.2$ m $\times$ $1.4$ m $\times$ $1$ m. A cylindrical closed tank has radius $0.7$ m and height $2$ m. (i) Find the capacity of each in litres. (ii) Which needs less sheet metal and by how much? (iii) A pump delivers 28 litres per minute; how long does it take to fill either tank? Use $π = \frac{22}{7}$ .

Try it first, then show the solutionSolution

(i) Cuboid: $2.2 \times 1.4 \times 1 = 3.08$ m $^{3} = 3080$ L. Cylinder: $\frac{22}{7} \times 0.49 \times 2 = 3.08$ m $^{3} = 3080$ L. (1)
(ii) Cuboid TSA $= 2 (3.08 + 1.4 + 2.2) = 13.36$ m $^{2}$ . Cylinder TSA $= 2 \times \frac{22}{7} \times 0.7 \times (0.7 + 2) = 4.4 \times 2.7 = 11.88$ m $^{2}$ . (1½)
The cylinder needs $13.36 - 11.88 = 1.48$ m $^{2}$ less metal. (½)
(iii) Time $= \frac{3080}{28} = 110$ minutes, that is 1 hour 50 minutes. (1)

Answer: (i) both 3080 L; (ii) the cylinder, by $1.48$ m $^{2}$ ; (iii) 110 minutes.

Where students lose marks in Cuboids, Cubes and Cylinders: Surface Area and Volume

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

Mistake: Using diameter as radius
Reading "diameter 0.7 m" as $r = 0.7$ .
Wrong working
CSA $= 2 π (0.7) (1.4)$ .
Right working
$r = 0.35$ m, so CSA $= 2 π (0.35) (1.4) = 3.08$ m $^{2}$ .
Do this: Underline "diameter" and halve it immediately.
Usually costs: The whole answer.
Mistake: Including a base that is not there
Using TSA for an open tank.
Wrong working
Sheet for open tank $= 2 π r (r + h)$ .
Right working
Open tank: $2 π r h + π r^{2}$ .
Do this: Count the faces explicitly.
Usually costs: 1 to 2 marks.
Mistake: Mixing units
Multiplying metres with centimetres.
Wrong working
$2.2 m \times 140 cm \times 1 m$ .
Right working
Convert to metres: $2.2 \times 1.4 \times 1 = 3.08$ m $^{3}$ .
Do this: Convert first, then multiply.
Usually costs: The whole answer.

Exam tips

Cancel the 7 in $\frac{22}{7}$ before multiplying large numbers.
Always state whether the surface is open, closed or lateral.
Convert litres, cm $^{3}$ and m $^{3}$ carefully: 1 m $^{3}$ is 1000 litres.
Write the formula, substitution and unit on separate lines for method marks.

Practice questions on Cuboids, Cubes and Cylinders: Surface Area and Volume

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
A solid cube has volume $343$ cm $^{3}$ . Its total surface area is
(a) 49 cm $^{2}$
(b) 196 cm $^{2}$
(c) 294 cm $^{2}$
(d) 343 cm $^{2}$
Hint
Find the edge as the cube root of the volume.
Answer
(c) 294 cm $^{2}$ . The edge is $7$ cm, so TSA $= 6 \times 7^{2} = 294$ .
Q2 · 3 marks · Short answerBoard level
A cylinder of height 10 cm has volume $1540$ cm $^{3}$ . Find its radius and its curved surface area. Use $π = \frac{22}{7}$ .
Hint
Use $π r^{2} h = 1540$ first.
Answer
$\frac{22}{7} \times r^{2} \times 10 = 1540$ gives $r^{2} = 49$ and $r = 7$ cm. CSA $= 2 \times \frac{22}{7} \times 7 \times 10 = 440$ cm $^{2}$ .
Q3 · 3 marks · Short answerHarder board level
The edges of a cuboid are in the ratio $4 : 3 : 2$ and its volume is $3000$ cm $^{3}$ . Find its edges and total surface area.
Hint
Let the edges be $4 x$ , $3 x$ , $2 x$ .
Answer
$24 x^{3} = 3000$ , so $x^{3} = 125$ and $x = 5$ . Edges: $20$ cm, $15$ cm, $10$ cm. TSA $= 2 (20 \times 15 + 15 \times 10 + 10 \times 20) = 2 (300 + 150 + 200) = 1300$ cm $^{2}$ .
Q4 · 5 marks · Long answerHarder board level
A cylindrical tank open at the top has radius 3.5 m and height 4 m. Find (i) the area of sheet needed to make it, (ii) the cost of the sheet at 120 rupees per square metre, (iii) its capacity in litres. Use $π = \frac{22}{7}$ .
Hint
Open at the top: one circular base only.
Answer
(i) Area $= 2 π r h + π r^{2} = π r (2 h + r) = \frac{22}{7} \times 3.5 \times (8 + 3.5) = 11 \times 11.5 = 126.5$ m $^{2}$ . (ii) Cost $= 126.5 \times 120 = 15180$ rupees. (iii) Volume $= \frac{22}{7} \times 12.25 \times 4 = 154$ m $^{3} = 154000$ litres.
Q5 · 4 marks · Case-basedHOTS / competency
Case: a garden roller is 1.4 m long and has a diameter of 0.7 m. (i) Find the area it covers in one full turn. (ii) How many turns does it make to level a ground of 616 m $^{2}$ ? (iii) If it makes 20 turns per minute, how long does the job take? (iv) If a roller twice as wide in radius (same length) were used, how many turns are needed? Use $π = \frac{22}{7}$ .
Hint
One turn covers the CSA; radius is half of the diameter.
Answer
(i) $r = 0.35$ m: CSA $= 2 \times \frac{22}{7} \times 0.35 \times 1.4 = 3.08$ m $^{2}$ . (ii) $\frac{616}{3.08} = 200$ turns. (iii) $\frac{200}{20} = 10$ minutes. (iv) The CSA doubles to $6.16$ m $^{2}$ , so the turns needed halve to $\frac{616}{6.16} = 100$ .

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.Curved surface area of a cylinder?Show answer
$2 π r h$ .
2.Volume of a cylinder?Show answer
$π r^{2} h$ .
3.Diagonal of a cuboid?Show answer
$l^{2} + b^{2} + h^{2}$ .
4.TSA of a cube?Show answer
$6 a^{2}$ .
5.1 cubic metre in litres?Show answer
1000 litres.

Cuboids, Cubes and Cylinders: Surface Area and Volume: frequently asked questions

Why is the curved surface area of a cylinder $2\pi rh$?

Unrolled, the curved surface is a rectangle whose length is the circumference 2\pi r and whose breadth is the height h.

Which value of pi should I use?

Use \frac{22}{7} unless the question gives another value, and state it in the first line.

Do two cylinders with the same curved surface area have the same volume?

No. As the rolled-sheet example shows, the same sheet can give different volumes.