Surface Area and Volume of Cylinder — CBSE Class 9 & 10 Explained
Volume Equals Base Area Multiplied by Height. Visualise the Cylinder Before Applying V = πr²h.
By Dhruv Chandravanshi | Class 9 & 10 Mathematics | CBSE
Have you ever wondered how a factory knows exactly how much metal sheet goes into making one tin can?
In this article, you will learn:
- What a cylinder actually is before formulas appear
- Why surface area and volume answer completely different questions about the same shape
- What CSA, TSA, and volume mean and when each formula should be used
- What CBSE Class 9 and 10 expect in mensuration problems and the mistakes that usually cost marks
By the end, you will see that cylinder formulas are not random equations to memorise.
Each formula describes something you can physically imagine and observe.
A Simple Observation from a Lassi Glass
Every morning, Arjun’s mother makes lassi in a tall steel glass commonly used in homes and tea stalls across Bihar.
Arjun examines the shape carefully.
It has:
- a circular base at the bottom
- another identical circle at the top
- a curved side joining both circles
He notices something interesting.
If the curved side were cut vertically and opened flat, it would become a rectangle.
So a cylinder is really made from:
- two circular surfaces
- one curved rectangular surface
Then Arjun asks two different questions:
- How much lassi can the glass hold?
- How much steel sheet is needed to make the glass?
Both questions involve the same object.
But they measure completely different things.
Surface Area and Volume Are Different Ideas
Surface Area
Surface area measures the outer covering of an object.
Imagine opening the cylinder and laying its surfaces flat.
You would get:
- one rectangle from the curved side
- two circles from the top and bottom
Adding these areas gives the total surface area.
Volume
Volume measures the space inside the cylinder.
Imagine filling the cylinder with many thin circular layers stacked one above another.
Each circular layer has area:
πr²
When these layers are stacked to height h, the total volume becomes:
V = πr²h
Important Terms Related to a Cylinder
A cylinder is a three-dimensional solid with two identical circular bases connected by a curved surface.
Important symbols:
- r → radius of the circular base
- h → height of the cylinder
- π → mathematical constant approximately equal to 22/7 or 3.14
Three Important Cylinder Formulas
Curved Surface Area (CSA)
This measures only the curved side of the cylinder.
CSA = 2πrh
Total Surface Area (TSA)
This includes:
- the curved surface
- the top circular base
- the bottom circular base
TSA = 2πrh + 2πr²
Another common form:
TSA = 2πr(h + r)
Volume
This measures the capacity inside the cylinder.
V = πr²h
How These Formulas Are Derived
Deriving the Curved Surface Area
Cut the cylinder vertically and open the curved surface.
The curved surface becomes a rectangle.
Rectangle dimensions:
- Height = h
- Length = circumference of base = 2πr
Area of rectangle:
CSA = 2πr × h
Therefore:
CSA = 2πrh
Deriving the Total Surface Area
The cylinder also contains two circular ends.
Area of one circle:
πr²
Area of two circles:
2πr²
Add this to the curved surface:
TSA = 2πrh + 2πr²
Deriving the Volume Formula
The cylinder can be imagined as stacked circular layers.
Each layer has area:
πr²
Multiplying by height gives:
V = πr²h
Choosing the Correct Formula
| Question Type | Formula Used |
|---|---|
| Wrapper, label, curved side only | CSA |
| Entire outer surface including top and bottom | TSA |
| Capacity, storage, liquid inside | Volume |
Worked Examples
Example 1: Curved Surface Area of a Tin Can
Radius = 7 cm
Height = 15 cm
CSA = 2πrh
Substitute values:
CSA = 2 × (22/7) × 7 × 15
CSA = 660 cm²
Example 2: Total Surface Area of a Water Tank
Radius = 3.5 m
Height = 10 m
TSA = 2πr(h + r)
Substitute values:
TSA = 2 × (22/7) × 3.5 × (10 + 3.5)
TSA = 297 m²
Example 3: Volume of a Lassi Glass
Radius = 4 cm
Height = 12 cm
V = πr²h
Substitute values:
V = (22/7) × 16 × 12
V ≈ 603 cm³
Since:
1 cm³ = 1 mL
the glass holds approximately 603 millilitres of liquid.
Example 4: Finding Height from Volume
Volume = 1540 cm³
Radius = 7 cm
V = πr²h
Substitute values:
1540 = (22/7) × 49 × h
1540 = 154h
h = 10 cm
Real-Life Applications of Cylinders
Water Pipes
Engineers calculate water flow and storage using cylinder volume formulas.
Tin Cans and Containers
Factories use surface area formulas to determine metal sheet requirements.
Storage Tanks
The capacity of cylindrical tanks and wells is calculated using:
V = πr²h
Common Mistakes Students Make
Mistake 1: Confusing CSA and TSA
CSA includes only the curved surface.
TSA includes:
- curved surface
- top base
- bottom base
Mistake 2: Mixing Area and Volume Formulas
Surface area formulas involve:
2πr
Volume formulas involve:
πr²h
Mixing them gives incorrect answers and wrong units.
Mistake 3: Using Diameter Instead of Radius
Many questions provide diameter directly.
Always convert first:
r = d ÷ 2
Mistake 4: Using Incorrect Value of π
Questions usually specify:
- π = 22/7
- π = 3.14
Use the value given in the question.
Important Unit Conversions
- 1 m = 100 cm
- 1 m² = 10,000 cm²
- 1 m³ = 10,00,000 cm³
- 1 litre = 1000 cm³
The Core Idea in Simple Words
A cylinder has:
- two circular bases
- one curved surface connecting them
The formulas come directly from these observations:
CSA = 2πrh
TSA = 2πrh + 2πr²
V = πr²h
Surface area measures covering.
Volume measures capacity.
Practice Questions
- A cylindrical water tank has radius 1.4 m and height 5 m. Find its CSA, TSA, and volume.
- A cylindrical tin can has radius 7 cm and curved surface area 440 cm². Find its height.
- A cylindrical well has diameter 140 cm and depth 20 m. Find the volume of earth removed.
- A hollow pipe has inner radius 4 cm, outer radius 4.5 cm, and length 21 m. Find the volume of material used.
- Compare the volume of a cylinder and a cone having the same radius and height.
Frequently Asked Questions
When should CSA be used?
Use CSA when only the curved surface is involved.
When should TSA be used?
Use TSA when the entire outer surface including top and bottom is required.
Why does volume contain r²?
Because volume uses the area of the circular base:
πr²
How do we convert litres into cubic centimetres?
1 litre = 1000 cm³
What happens if the radius doubles?
Since volume depends on r²:
doubling the radius increases volume four times.
Related Topics
- Surface Area and Volume of a Cone
- Surface Area and Volume of a Sphere
- Areas Related to Circles
This article follows the CBSE Class 9 and Class 10 Mathematics syllabus (Surface Areas and Volumes) and aligns with NCERT concepts and standard CBSE examination patterns.
About the Author
Dhruv Chandravanshi writes about physical and logical systems, exploring how mathematical structures describe the behaviour of the real world.
