Volume of Cylinder
A cylinder is a three-dimensional solid with two parallel circular bases connected by a curved surface. Common examples include cans, pipes, water tanks, and pillars.
The volume of a cylinder measures the amount of space enclosed inside it, or the capacity of the cylinder. If you want to know how much water a cylindrical tank can hold, you need to find its volume.
In Class 8 Mathematics (NCERT), the volume of a cylinder is studied in the chapter Mensuration. The formula uses two measurements: the radius (r) of the circular base and the height (h) of the cylinder.
The volume of a cylinder is derived from the basic principle: Volume = Area of base × Height. Since the base is a circle with area πr², the volume becomes πr²h.
What is Volume of Cylinder?
Definition: The volume of a cylinder is the amount of three-dimensional space enclosed within it. It equals the product of the area of its circular base and its height.
Key terms:
- Cylinder: A 3D solid with two parallel, congruent circular bases connected by a curved surface.
- Radius (r): The radius of the circular base.
- Height (h): The perpendicular distance between the two circular bases.
- Volume: The space enclosed inside the cylinder, measured in cubic units (cm³, m³, litres, etc.).
- Right circular cylinder: A cylinder where the axis is perpendicular to the base. This is the type studied in Class 8.
Important relationships:
- Diameter = 2r. If the diameter is given, divide by 2 to get the radius.
- 1 litre = 1,000 cm³. This conversion is needed for capacity problems.
- 1 m³ = 1,000 litres.
Volume of Cylinder Formula
Volume of Cylinder:
V = πr²h
Where:
- V = volume of the cylinder
- π = 22/7 or 3.14159...
- r = radius of the circular base
- h = height of the cylinder
If the diameter (d) is given:
V = π(d/2)²h = πd²h/4
Related formulas:
- To find height: h = V / (πr²)
- To find radius: r = √(V / πh)
- Capacity in litres: Volume in cm³ ÷ 1000
Derivation and Proof
Deriving the volume formula:
Principle: The volume of any prism or cylinder = Area of base × Height.
- The base of a cylinder is a circle.
- Area of the circular base = πr².
- The cylinder extends this base upward to a height h.
- Volume = Area of base × Height = πr² × h = πr²h.
Visual understanding:
- Imagine stacking many thin circular discs (like coins) one on top of another.
- Each disc has area πr² and very small thickness.
- When you stack them to a total height h, you get a cylinder.
- Total volume = sum of all disc volumes = πr² × h.
Why the formula works for any height:
- A cylinder has a uniform cross-section — every horizontal slice is the same circle.
- This means the volume scales linearly with height: double the height → double the volume.
- The formula πr²h captures this relationship perfectly.
Relationship with cuboid:
- For a cuboid: Volume = l × b × h (base area × height).
- For a cylinder: Volume = πr² × h (base area × height).
- The same principle applies — only the shape of the base changes.
Types and Properties
Types of volume of cylinder problems:
1. Finding volume given r and h:
- Directly apply V = πr²h.
- Most basic type of problem.
2. Finding volume given diameter and height:
- First find r = d/2, then apply V = πr²h.
- Common mistake: using diameter instead of radius in the formula.
3. Finding height given volume and radius:
- Rearrange: h = V / (πr²).
4. Finding radius given volume and height:
- Rearrange: r² = V / (πh), then r = √(V/πh).
5. Capacity problems (litres):
- Find volume in cm³, then convert: litres = cm³ ÷ 1000.
- Used for tanks, containers, glasses, etc.
6. Comparing volumes:
- Compare volumes of two cylinders with different dimensions.
- Useful for understanding how changes in r or h affect volume.
7. Hollow cylinder:
- Volume = π(R² − r²)h, where R = outer radius, r = inner radius.
- Used for pipes, tubes, and rings.
Solved Examples
Example 1: Example 1: Basic volume calculation
Problem: Find the volume of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)
Solution:
Given:
- r = 7 cm, h = 10 cm
Using the formula:
- V = πr²h
- V = (22/7) × 7² × 10
- V = (22/7) × 49 × 10
- V = 22 × 7 × 10
- V = 1,540 cm³
Answer: The volume is 1,540 cm³.
Example 2: Example 2: Volume given diameter
Problem: Find the volume of a cylinder with diameter 14 cm and height 20 cm.
Solution:
Given:
- Diameter = 14 cm, so r = 14/2 = 7 cm
- h = 20 cm
Using the formula:
- V = πr²h = (22/7) × 7² × 20
- V = (22/7) × 49 × 20
- V = 22 × 7 × 20
- V = 3,080 cm³
Answer: The volume is 3,080 cm³.
Example 3: Example 3: Capacity of a tank in litres
Problem: A cylindrical water tank has radius 35 cm and height 1 m. Find its capacity in litres.
Solution:
Given:
- r = 35 cm, h = 1 m = 100 cm
Step 1: Find volume in cm³:
- V = πr²h = (22/7) × 35² × 100
- V = (22/7) × 1225 × 100
- V = 22 × 175 × 100
- V = 3,85,000 cm³
Step 2: Convert to litres:
- Capacity = 3,85,000 ÷ 1000 = 385 litres
Answer: The tank can hold 385 litres of water.
Example 4: Example 4: Finding height from volume
Problem: A cylinder has volume 1,232 cm³ and radius 7 cm. Find its height.
Solution:
Given:
- V = 1,232 cm³, r = 7 cm, h = ?
Using the formula:
- V = πr²h
- 1,232 = (22/7) × 7² × h
- 1,232 = (22/7) × 49 × h
- 1,232 = 22 × 7 × h
- 1,232 = 154h
- h = 1,232/154 = 8 cm
Answer: The height is 8 cm.
Example 5: Example 5: Finding radius from volume
Problem: A cylinder has volume 2,464 cm³ and height 8 cm. Find the radius.
Solution:
Given:
- V = 2,464 cm³, h = 8 cm, r = ?
Using the formula:
- V = πr²h
- 2,464 = (22/7) × r² × 8
- 2,464 = (176/7) × r²
- r² = 2,464 × 7/176
- r² = 17,248/176
- r² = 98
- r = √98 = 7√2 ≈ 9.9 cm
Answer: The radius is approximately 9.9 cm (or exactly 7√2 cm).
Example 6: Example 6: Comparing two cylinders
Problem: Cylinder A has radius 7 cm and height 14 cm. Cylinder B has radius 14 cm and height 7 cm. Which has greater volume?
Solution:
Cylinder A:
- V₁ = πr²h = (22/7) × 7² × 14 = 22 × 7 × 14 = 2,156 cm³
Cylinder B:
- V₂ = πr²h = (22/7) × 14² × 7 = (22/7) × 196 × 7 = 22 × 196 = 4,312 cm³
Comparison: Cylinder B has double the volume of Cylinder A.
Reason: Volume depends on r² (square of radius) but only h (first power of height). Doubling the radius has a greater effect than doubling the height.
Answer: Cylinder B has greater volume (4,312 cm³ vs 2,156 cm³).
Example 7: Example 7: Hollow cylinder (pipe)
Problem: A cylindrical pipe has outer radius 10 cm, inner radius 8 cm, and length 35 cm. Find the volume of metal used.
Solution:
Given:
- R (outer) = 10 cm, r (inner) = 8 cm, h = 35 cm
Volume of metal = Volume of outer cylinder − Volume of inner cylinder:
- V = π(R² − r²)h
- V = (22/7)(10² − 8²) × 35
- V = (22/7)(100 − 64) × 35
- V = (22/7) × 36 × 35
- V = 22 × 36 × 5
- V = 3,960 cm³
Answer: The volume of metal is 3,960 cm³.
Example 8: Example 8: Water flow problem
Problem: Water flows through a cylindrical pipe of radius 3.5 cm at 2 m/s. How much water flows in 5 minutes? Give the answer in litres.
Solution:
Given:
- r = 3.5 cm, speed = 2 m/s = 200 cm/s, time = 5 min = 300 s
Step 1: Length of water column in 5 min:
- Length = speed × time = 200 × 300 = 60,000 cm
Step 2: Volume = πr²h:
- V = (22/7) × 3.5² × 60,000
- V = (22/7) × 12.25 × 60,000
- V = 22 × 1.75 × 60,000
- V = 23,10,000 cm³
Step 3: Convert to litres:
- V = 23,10,000 ÷ 1000 = 2,310 litres
Answer: 2,310 litres of water flows in 5 minutes.
Example 9: Example 9: Filling a tank
Problem: A cylindrical tank of radius 1.4 m and height 2 m is to be filled with water. If water costs Rs 5 per litre, find the cost to fill the tank.
Solution:
Given:
- r = 1.4 m = 140 cm, h = 2 m = 200 cm
Step 1: Volume:
- V = πr²h = (22/7) × 140² × 200
- V = (22/7) × 19,600 × 200
- V = 22 × 2,800 × 200
- V = 1,23,20,000 cm³
Step 2: In litres:
- V = 1,23,20,000 ÷ 1000 = 12,320 litres
Step 3: Cost = 12,320 × 5 = Rs 61,600
Answer: The cost to fill the tank is Rs 61,600.
Example 10: Example 10: Effect of doubling radius
Problem: A cylinder has radius r and height h. If the radius is doubled and the height is halved, how does the volume change?
Solution:
Original volume:
- V₁ = πr²h
New dimensions: radius = 2r, height = h/2
- V₂ = π(2r)²(h/2)
- V₂ = π × 4r² × h/2
- V₂ = 2πr²h
Ratio:
- V₂/V₁ = 2πr²h / πr²h = 2
Answer: The new volume is double the original volume.
Real-World Applications
Real-world applications of volume of cylinder:
- Water tanks: Calculating the capacity of cylindrical overhead tanks, underground tanks, and tanker trucks.
- Pipes and plumbing: Finding the volume of water flowing through cylindrical pipes per unit time.
- Containers: Determining how much a cylindrical can, jar, or bottle can hold (capacity in litres or ml).
- Construction: Calculating the volume of cylindrical concrete pillars, columns, and poles.
- Engineering: Designing pistons, hydraulic cylinders, and engine components.
- Agriculture: Estimating the volume of cylindrical grain silos and water wells.
- Cooking: Measuring the capacity of cylindrical vessels, glasses, and pots.
- Manufacturing: Calculating the amount of material needed to make cylindrical objects (hollow cylinder formula for pipes and tubes).
Key Points to Remember
- The volume of a cylinder = πr²h.
- π = 22/7 or 3.14159 (use whichever is specified in the problem).
- Always use radius, not diameter, in the formula. If diameter is given, divide by 2.
- Volume is measured in cubic units (cm³, m³).
- 1 litre = 1,000 cm³ and 1 m³ = 1,000 litres.
- Volume depends on r² — doubling the radius quadruples the volume (if h is unchanged).
- Volume depends linearly on h — doubling the height doubles the volume.
- For a hollow cylinder (pipe): V = π(R² − r²)h.
- All dimensions must be in the same unit before applying the formula.
- Volume formula is derived from: Volume = Area of base × Height.
Practice Problems
- Find the volume of a cylinder with radius 10.5 cm and height 15 cm. (Use π = 22/7)
- A cylindrical vessel has diameter 28 cm and height 30 cm. Find its capacity in litres.
- The volume of a cylinder is 6,160 cm³ and its radius is 14 cm. Find the height.
- A cylinder has volume 3,850 cm³ and height 10 cm. Find the radius.
- A cylindrical pipe has outer diameter 8 cm, inner diameter 6 cm, and length 50 cm. Find the volume of the pipe material.
- Water flows through a pipe of radius 7 cm at 5 m/s. How many litres flow in 2 minutes?
- A cylinder has radius 7 cm and height 20 cm. If the radius is tripled, what is the new volume?
- A cylindrical tank (radius 2.1 m, height 3 m) is to be filled. If water costs Rs 3 per litre, find the total cost.
Frequently Asked Questions
Q1. What is the formula for volume of a cylinder?
The volume of a cylinder is V = πr²h, where r is the radius of the circular base and h is the height of the cylinder.
Q2. What is the difference between volume and capacity?
Volume is the total space a solid occupies (in cm³ or m³). Capacity is the amount of liquid a container can hold (in litres or ml). Numerically, 1 litre = 1,000 cm³.
Q3. What happens to volume if the radius is doubled?
If the radius is doubled (keeping height the same), the volume becomes 4 times the original. This is because volume depends on r² — (2r)² = 4r².
Q4. What happens to volume if the height is doubled?
If the height is doubled (keeping radius the same), the volume also doubles. Volume is directly proportional to height.
Q5. How do you find the volume of a hollow cylinder?
Volume of hollow cylinder = π(R² − r²)h, where R is the outer radius, r is the inner radius, and h is the height. This gives the volume of the material (wall) of the cylinder.
Q6. How do you convert cm³ to litres?
Divide by 1,000. 1 litre = 1,000 cm³. For example, 5,000 cm³ = 5 litres.
Q7. Can volume be calculated using diameter instead of radius?
Yes. If the diameter d is given, use r = d/2. The formula becomes V = π(d/2)²h = πd²h/4.
Q8. Why is the volume formula πr²h?
The volume of any shape with a uniform cross-section equals the area of the cross-section times the height. For a cylinder, the cross-section is a circle (area = πr²), so volume = πr² × h.
Q9. What value of π should be used?
Use π = 22/7 when the radius or height is a multiple of 7 (for easier calculation). Use π = 3.14 in other cases. Follow the instruction given in the problem.
Q10. What is the volume of a cylinder with radius 0 or height 0?
If either r = 0 or h = 0, the volume is 0. A cylinder must have both a non-zero radius and non-zero height to enclose any space.
