MathsICSE Class 8 · Chapter 21

Surface Area and Volume ICSE Class 8: Cube, Cuboid and Cylinder

Updated 15 October 2026 · Fact-checked

Surface area is the total area of the faces of a solid; volume is the space it fills. In ICSE Class 8, a cuboid has surface area 2(lb + bh + lh) and volume lbh, a cube 6a² and a³, and a cylinder curved surface 2πrh, total surface 2πr(r + h) and volume πr²h, with 1 litre = 1000 cm³.

What Surface Area and Volume is about

A solid has two important measures:

surface area: the total area of all its outer surfaces (how much paper wraps it, how much paint covers it), in cm² or m²;
volume: the amount of space it takes up (how much it holds), in cm³ or m³.

The CISCE Class VIII curriculum asks for:

surface area of a cube, cuboid and cylinder,
the idea of total surface area (TSA) and curved (or lateral) surface area (CSA/LSA),
the concept of volume using unit cubes, and the volume of a cube, cuboid and cylinder,
volume and capacity: $1$ litre $= 1000$ cm³ and $1$ m³ $= 1000$ litres.

A useful picture: volume counts unit cubes (1 cm × 1 cm × 1 cm) that fill the solid. A cuboid 4 cm × 3 cm × 2 cm holds $4 \times 3 \times 2 = 24$ unit cubes, so its volume is 24 cm³.

In a school exam you will compute areas and volumes, work backwards to a missing dimension, convert to litres, and solve costing problems (painting, wrapping, filling tanks).

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Surface Area and Volume sits in the Mensuration theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Surface area of a cuboid

A cuboid with length $l$ , breadth $b$ and height $h$ has six rectangular faces in three matching pairs:
$TSA = 2 (l b + bh + l h)$ .

The lateral surface area (the four walls, without top and bottom) is
$LSA = 2 h (l + b)$ .

LSA is used for the walls of a room (painting), or a box with no lid or base.

Worked example

Find the TSA and LSA of a cuboid 10 cm × 8 cm × 5 cm.

Try it first, then show the solutionSolution

TSA $= 2 (80 + 40 + 50) = 2 \times 170 = 340$ cm².
LSA $= 2 \times 5 \times (10 + 8) = 180$ cm².

Answer: TSA 340 cm²; LSA 180 cm²

Surface area of a cube

A cube of edge $a$ has six equal square faces:

$TSA = 6 a^{2}$ ,
$LSA = 4 a^{2}$ (four side faces).

If you know the TSA, divide by 6 to get the area of one face, then take the square root for the edge.

Worked example

Find the TSA and LSA of a cube of edge 6 cm.

Try it first, then show the solutionSolution

TSA $= 6 \times 36 = 216$ cm².
LSA $= 4 \times 36 = 144$ cm².

Answer: TSA 216 cm²; LSA 144 cm²

Surface area of a cylinder

Unroll the curved surface of a cylinder (radius $r$ , height $h$ ): it becomes a rectangle with length equal to the circumference $2 π r$ and width $h$ .

Curved surface area: $CSA = 2 π r h$ .
Total surface area (closed cylinder) $=$ CSA $+$ two circular ends $= 2 π r h + 2 π r^{2} = 2 π r (r + h)$ .
An open-topped cylinder (like a glass) has CSA $+ π r^{2}$ .

Take $π = \frac{22}{7}$ unless told otherwise.

Worked example

A closed cylinder has radius 7 cm and height 10 cm. Find its CSA and TSA.

Try it first, then show the solutionSolution

CSA $= 2 \times \frac{22}{7} \times 7 \times 10 = 440$ cm².
Two ends $= 2 \times \frac{22}{7} \times 7 \times 7 = 308$ cm².
TSA $= 440 + 308 = 748$ cm² (or $2 \times \frac{22}{7} \times 7 \times 17 = 748$ ).

Answer: CSA 440 cm²; TSA 748 cm²

Volume of a cuboid and a cube

Volume counts the unit cubes that fill a solid. A cuboid holds $l$ cubes along, $b$ across and $h$ layers up:

cuboid: $V = l \times b \times h$ ,
cube: $V = a^{3}$ .

Also $V = area of base \times height$ .

All three lengths must be in the same unit before multiplying.

Worked example

Find the volume of a cuboid 10 cm × 8 cm × 5 cm, and say how many 1 cm unit cubes fill it.

Try it first, then show the solutionSolution

$V = 10 \times 8 \times 5 = 400$ cm³.
So 400 unit cubes fill it: 10 along, 8 across, 5 layers.

Answer: 400 cm³; 400 unit cubes

Volume of a cylinder

A cylinder is a stack of circular layers, so
$V = area of base \times height = π r^{2} h$ .

If the diameter is given, halve it first.

Worked example

Find the volume of a cylinder of radius 7 cm and height 10 cm, in cm³ and in litres.

Try it first, then show the solutionSolution

$V = \frac{22}{7} \times 7 \times 7 \times 10 = 1540$ cm³.
$1540 \div 1000 = 1.54$ litres.

Answer: 1540 cm³ = 1.54 L

Volume and capacity

Capacity is how much a container can hold, usually in litres (L) or millilitres (mL):

$1$ cm³ $= 1$ mL,
$1000$ cm³ $= 1$ L,
$1$ m³ $= 1000$ L.

To find how many small objects fit in a box, check how many fit along each edge and multiply; dividing volumes works only when the edges divide exactly.

Worked example

A tank measures 2 m × 1.5 m × 1 m. How many litres does it hold?

Try it first, then show the solutionSolution

Volume $= 2 \times 1.5 \times 1 = 3$ m³.
Capacity $= 3 \times 1000 = 3000$ litres.

Answer: 3000 litres

Maths in real life: Surface Area and Volume

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.

Cylinder volume

Suppose a water bottle is a cylinder of radius 3.5 cm and height 20 cm.

How much water does it hold?

Try it first, then show the solutionSolution

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

Answer: 770 mL

Memory hook: Base area × height.

Cuboid surface area

Imagine gift-wrapping a box 30 cm × 20 cm × 10 cm with no overlaps.

How much paper covers it exactly?

Try it first, then show the solutionSolution

TSA $= 2 (600 + 200 + 300)$ .
$= 2200$ cm².

Answer: 2200 cm²

Memory hook: Wrapping paper = total surface area.

Capacity

Suppose a 1-litre milk carton has a base 10 cm × 5 cm.

How tall must it be to hold exactly 1 litre?

Try it first, then show the solutionSolution

1 L $= 1000$ cm³ $= 10 \times 5 \times h$ .
$h = 1000 \div 50 = 20$ cm.

Answer: 20 cm

Memory hook: Volume ÷ base area = height.

Topics in this chapter

Surface Area and Volume formula sheet

Cuboid TSA: $2 (l b + bh + l h)$; LSA = 2h(l + b).
Cube TSA: $6 a^{2}$; LSA = 4a².
Cylinder CSA: $2 π r h$; Unrolled rectangle.
Cylinder TSA: $2 π r (r + h)$; Closed cylinder.
Volumes: $V = l bh; V = a^{3}; V = π r^{2} h$; Cuboid, cube, cylinder.
Capacity: $1 L = 1000 cm^{3}$; 1 m³ = 1000 L.

How the school exam asks Surface Area and Volume

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Surface area	Find the TSA or LSA of a box or the CSA/TSA of a tin.	Choose the right formula: total, lateral or curved.
Volume	Find the volume of a cuboid, cube or cylinder; convert to litres.	lbh, a³, πr²h; then ÷ 1000 for litres.
Missing dimension	Find the edge of a cube from its TSA or volume; the height of a cylinder from its CSA.	Rearrange the formula.
Cost problems	Cost of painting walls, wrapping a box, filling a tank.	Area × rate for surfaces; volume × rate for contents.
Packing and melting	How many bricks make a wall? A cuboid is melted into a cube.	Volumes are equal when melted; count along edges when packing.

Tricks to remember Surface Area and Volume

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

Area wraps, volume fills
Surface area = wrapping paper; volume = what goes inside.
Use it when: Choosing the right measure.
Three pairs of faces
A cuboid's TSA is 2 × (each different face added once).
Use it when: 2(lb + bh + lh).
Unroll the can
The curved surface of a cylinder unrolls into a rectangle 2πr long and h high.
Use it when: CSA = 2πrh.
1000 cm³ makes a litre
A 10 cm cube holds exactly 1 litre.
Use it when: Converting to litres.

Surface Area and Volume: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Volume of a cube · 1 mark · MCQExam level

The volume of a cube of edge 5 cm is:
(a) 25 cm³
(b) 125 cm³
(c) 150 cm³
(d) 60 cm³

Try it first, then show the solutionSolution

$V = a^{3} = 5^{3}$ .
$= 125$ cm³ (150 is its TSA).

Answer: (b)

Example 2: Cuboid box · 3 marks · Short answerExam level

A box measures 12 cm × 8 cm × 6 cm. Find its total surface area and volume.

Try it first, then show the solutionSolution

TSA $= 2 (96 + 48 + 72) = 2 \times 216$ .
$= 432$ cm².
Volume $= 12 \times 8 \times 6 = 576$ cm³.

Answer: 432 cm²; 576 cm³

Example 3: A small cylinder · 3 marks · Short answerExam level

A cylinder has radius 3.5 cm and height 10 cm. Find its curved surface area and volume.

Try it first, then show the solutionSolution

CSA $= 2 \times \frac{22}{7} \times 3.5 \times 10 = 220$ cm².
Volume $= \frac{22}{7} \times 3.5 \times 3.5 \times 10$ .
$= 385$ cm³.

Answer: CSA 220 cm²; volume 385 cm³

Example 4: Edge from volume · 3 marks · Short answerHarder exam level

The volume of a cube is 512 cm³. Find its total surface area.

Try it first, then show the solutionSolution

$a^{3} = 512 = 8^{3}$ , so $a = 8$ cm.
TSA $= 6 \times 64$ .
$= 384$ cm².

Answer: 384 cm²

Example 5: Painting a room · 4 marks · Long answerHarder exam level

A room is 6 m long, 4 m wide and 3 m high. Its four walls and ceiling are to be painted, leaving out a door of area 2 m² and a window of area 1.5 m². Find the area painted and the cost at ₹20 per m².

Try it first, then show the solutionSolution

Walls (LSA) $= 2 \times 3 \times (6 + 4) = 60$ m².
Ceiling $= 6 \times 4 = 24$ m²; total $= 84$ m².
Less door and window: $84 - 2 - 1.5 = 80.5$ m².
Cost $= 80.5 \times 20 = ₹1610$ .

Answer: 80.5 m²; ₹1,610

Example 6: A cylindrical tank · 4 marks · Long answerHarder exam level

A cylindrical water tank has diameter 1.4 m and height 2 m. How many litres of water can it hold?

Try it first, then show the solutionSolution

$r = 0.7$ m.
$V = \frac{22}{7} \times 0.7 \times 0.7 \times 2 = 3.08$ m³.
$1$ m³ $= 1000$ L.
Capacity $= 3080$ litres.

Answer: 3080 litres

Example 7: Packing cubes · 5 marks · Long answerHOTS / competency

(i) How many cubes of edge 2 cm can be packed into a box 10 cm × 8 cm × 6 cm? (ii) How many fit into a box 10 cm × 9 cm × 6 cm? Explain why dividing the volumes gives the wrong answer in (ii).

Try it first, then show the solutionSolution

(i) Along each edge: $10 \div 2 = 5$ , $8 \div 2 = 4$ , $6 \div 2 = 3$ ; total $5 \times 4 \times 3 = 60$ .
Check: $\frac{480}{8} = 60$ ✓, because every edge divides exactly.
(ii) Along 9 cm only 4 cubes fit (8 cm), leaving a 1 cm gap; total $5 \times 4 \times 3 = 60$ .
Dividing volumes gives $\frac{540}{8} = 67.5$ , which is impossible.
The 1 cm gap cannot hold a 2 cm cube, so only whole cubes along each edge count.

Answer: (i) 60 (ii) 60; volume division ignores the gap

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Choosing a juice pack

Suppose a juice company compares two closed containers, both 12 cm tall: a cuboid pack with base 7 cm × 5 cm, and a cylindrical can of radius 3.5 cm. Take $π = \frac{22}{7}$ .

(i) · 3 marks
Find the volume of each container.
Answer
Answer: Pack 420 cm³; can 462 cm³
Pack: $7 \times 5 \times 12 = 420$ cm³. Can: $\frac{22}{7} \times 3.5 \times 3.5 \times 12 = 462$ cm³.
(ii) · 3 marks
Find the material (total surface area) needed for each.
Answer
Answer: Pack 358 cm²; can 341 cm²
Pack: $2 (35 + 60 + 84) = 358$ cm². Can: $2 \times \frac{22}{7} \times 3.5 \times (3.5 + 12) = 22 \times 15.5 = 341$ cm².
(iii) · 4 marks
Which container holds more juice for each cm² of material? Show your working.
Answer
Answer: The can
Pack: $420 \div 358 \approx 1.17$ cm³ per cm². Can: $462 \div 341 \approx 1.35$ cm³ per cm². The can holds more and uses less material.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Find the total surface area and volume of a cube of edge 9 cm. (3 marks)

Step	Marks
TSA $= 6 a^{2} = 6 \times 81$	1
$= 486$ cm²	1
Volume $= 9^{3} = 729$ cm³	1

A closed cylinder has radius 10.5 cm and height 14 cm. Find its CSA, TSA and volume, and its capacity in litres. (4 marks)

Step	Marks
CSA $= 2 \times \frac{22}{7} \times 10.5 \times 14 = 924$ cm²	1
TSA $= 924 + 2 \times \frac{22}{7} \times 10. 5^{2} = 924 + 693 = 1617$ cm²	1
Volume $= \frac{22}{7} \times 10. 5^{2} \times 14 = 4851$ cm³	1
Capacity $= 4.851$ L	1

An open-topped glass aquarium is 80 cm long, 40 cm wide and 50 cm high. Find (i) the area of glass used (ii) its capacity in litres (iii) the litres of water when it is filled to a depth of 40 cm. (5 marks)

Step	Marks
Base $= 80 \times 40 = 3200$ cm²	1
Four sides $= 2 (80 \times 50) + 2 (40 \times 50) = 8000 + 4000 = 12000$ cm²	1
(i) Glass $= 15200$ cm² (no top)	1
(ii) Volume $= 80 \times 40 \times 50 = 160000$ cm³ $= 160$ L	1
(iii) $80 \times 40 \times 40 = 128000$ cm³ $= 128$ L	1

Where students lose marks in Surface Area and Volume

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

Mistake: Mixing up TSA and volume units
Areas are in square units, volumes in cubic units.
Wrong working
Volume $= 576$ cm²
Right working
Volume $= 576$ cm³
Do this: Area: cm². Volume: cm³.
Usually costs: 1 mark.
Mistake: Using the diameter as the radius
Formulas use r.
Wrong working
Diameter 1.4 m: $V = \frac{22}{7} \times 1. 4^{2} \times 2$
Right working
$r = 0.7$ m: $V = 3.08$ m³
Do this: Halve the diameter first.
Usually costs: The whole answer.
Mistake: Including the top for an open box or tank
An open container has no lid.
Do this: Read 'open', 'without lid', 'walls only' carefully and leave out those faces.
Usually costs: 1 to 2 marks.
Mistake: Mixing units
Multiplying metres by centimetres gives nonsense.
Do this: Convert all lengths to one unit before multiplying.
Usually costs: The answer.
Mistake: Wrong litre conversion
1 m³ is 1000 L, not 100 L; 1 L is 1000 cm³.
Do this: Remember the 10 cm cube = 1 L.
Usually costs: 1 mark.

Practice questions on 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 · MCQExam level
1 litre is equal to:
(a) 100 cm³
(b) 1000 cm³
(c) 10 cm³
(d) 1 m³
Hint
A 10 cm cube.
Answer
(b)
Q2 · 2 marks · Short answerExam level
Find the lateral surface area of a cuboid 10 cm × 6 cm × 4 cm (height 4 cm).
Hint
2h(l + b).
Answer
$2 \times 4 \times 16 = 128$ cm².
Q3 · 2 marks · Short answerExam level
Find the volume of a cylinder of radius 7 cm and height 5 cm.
Hint
πr²h.
Answer
$\frac{22}{7} \times 49 \times 5 = 770$ cm³.
Q4 · 3 marks · Short answerHarder exam level
The total surface area of a cube is 294 cm². Find its volume.
Hint
One face $= 294 \div 6$ .
Answer
Face $= 49$ cm², edge 7 cm, volume $= 343$ cm³.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): If the edge of a cube is doubled, its volume becomes 8 times.
Reason (R): The volume of a cube is $a^{3}$ .
(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
$(2 a)^{3}$ .
Answer
(c)
Q6 · 2 marks · Short answerHarder exam level
A water tank is 2.5 m × 1.6 m × 1 m. How many litres does it hold?
Hint
m³ × 1000.
Answer
$4$ m³ $= 4000$ L.
Q7 · 3 marks · Short answerHarder exam level
The curved surface area of a cylinder is 880 cm² and its radius is 14 cm. Find its height.
Hint
$h = \frac{CSA}{2 π r}$ .
Answer
$2 \times \frac{22}{7} \times 14 = 88$ ; $h = 880 \div 88 = 10$ cm.
Q8 · 3 marks · Short answerHarder exam level
How many bricks 20 cm × 10 cm × 7.5 cm are needed for a wall 10 m long, 30 cm thick and 2 m high (ignore mortar)?
Hint
Change all to metres.
Answer
Wall $= 10 \times 0.3 \times 2 = 6$ m³; brick $= 0.2 \times 0.1 \times 0.075 = 0.0015$ m³; bricks $= 6 \div 0.0015 = 4000$ .
Q9 · 4 marks · Long answerHarder exam level
A closed tin can has radius 7 cm and height 20 cm. Find the metal needed to make it and its capacity in litres.
Hint
TSA = 2πr(r + h).
Answer
TSA $= 2 \times \frac{22}{7} \times 7 \times 27 = 1188$ cm². Volume $= \frac{22}{7} \times 49 \times 20 = 3080$ cm³ $= 3.08$ L.
Q10 · 2 marks · Short answerHarder exam level
A cuboid has volume 1200 cm³, length 15 cm and breadth 10 cm. Find its height.
Hint
$h = V \div (l b)$ .
Answer
$1200 \div 150 = 8$ cm.
Q11 · 4 marks · Long answerHOTS / competency
During a storm, 2 cm of rain falls on a flat roof 10 m × 8 m. All of it drains into a cylindrical tank of radius 1 m. By how much does the water level rise? ( $π = 3.14$ )
Hint
Volume of rain = 10 × 8 × 0.02 m³.
Answer
Rain $= 1.6$ m³. Rise $= \frac{1.6}{3.14 \times 1 ^{2}} \approx 0.51$ m (about 51 cm).
Q12 · 4 marks · Long answerHOTS / competency
A metal cuboid 16 cm × 8 cm × 4 cm is melted and recast into a cube. Find the edge of the cube and compare the surface areas of the cuboid and the cube.
Hint
Volume is unchanged.
Answer
Volume $= 512$ cm³, so the cube's edge is 8 cm. Cube TSA $= 384$ cm²; cuboid TSA $= 2 (128 + 32 + 64) = 448$ cm². The cube has 64 cm² less surface.

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.TSA of a cuboid?Show answer
2(lb + bh + lh).
2.Volume of a cylinder?Show answer
πr²h.
3.CSA of a cylinder?Show answer
2πrh.
4.How many cm³ make 1 litre?Show answer
1000.
5.How many litres in 1 m³?Show answer
1000.
6.Volume of a cube of edge 4 cm?Show answer
64 cm³.
7.LSA of a cube of edge 3 cm?Show answer
36 cm².
8.Which formula is used to paint the four walls of a room?Show answer
LSA = 2h(l + b).

Best order to study it

Revise areas of rectangles and circles, and the circumference.
Learn TSA and LSA of the cuboid and cube.
Learn CSA and TSA of the cylinder by unrolling it.
Learn volumes of cuboid, cube and cylinder with unit cubes.
Finish with capacity in litres and real-life costing and packing problems.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Did you choose total, lateral or curved surface area correctly?
Did you halve the diameter?
Are all lengths in the same unit?
Did you write cm² for areas and cm³ for volumes?
Did you convert to litres correctly?
For an open container, did you leave out the top?

Surface Area and Volume: frequently asked questions

What is the difference between TSA and LSA?

TSA counts every face. LSA counts only the side faces (for a cuboid, the four walls), leaving out the top and bottom.

Why is the CSA of a cylinder 2πrh?

If you cut and unroll the curved surface, you get a rectangle whose length is the circumference 2πr and whose width is the height h.

What is capacity?

The amount a container can hold, usually measured in litres or millilitres. 1 cm³ = 1 mL.

When a solid is melted and recast, what stays the same?

The volume stays the same (no metal is lost), but the surface area usually changes.

Is the curved surface of a cylinder a face?

It is a curved surface, not a flat face. That is why a cylinder is not a polyhedron, although we can still find its surface area.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII