Basic Volume of Cylinder Formula
The fundamental formula to find the volume of a cylinder is:
V = πr²h
Where:
- V = Volume
- π = Pi (approximately 3.14159 or 22/7)
- r = Radius of the base
- h = Height of the cylinder
Why this formula works: Volume equals the area of the circular base (πr²) multiplied by the height (h).
Cylinder Volume Formulas
Here’s a comprehensive table covering all variations of cylinder volume formulas students need:
| Formula Type |
Formula |
When to Use |
Variables |
| Basic Volume Formula |
V = πr²h |
Standard cylinder problems |
r = radius, h = height |
| Volume Using Diameter |
V = π(d/2)²h or V = πd²h/4 |
When diameter is given instead of radius |
d = diameter, h = height |
| Volume of Hollow Cylinder |
V = πh(R² – r²) |
For pipes, tubes, rings with thickness |
R = outer radius, r = inner radius, h = height |
| Volume Using Circumference |
V = C²h/4π |
When base circumference is known |
C = circumference, h = height |
| Volume in Terms of Base Area |
V = A × h |
When base area is already calculated |
A = base area, h = height |
| Volume Using Lateral Surface Area |
V = (LSA × r)/2 |
When lateral surface area is given |
LSA = lateral surface area, r = radius |
| Volume of Half Cylinder |
V = πr²h/2 |
For semi-cylindrical shapes |
r = radius, h = length |
| Volume Using Total Surface Area |
V = r(TSA – 2πr²)/2 |
When total surface area is known |
TSA = total surface area, r = radius |
Volume of Hollow Cylinder Formula
A hollow cylinder has thickness—like a pipe or a tube. It’s essentially one cylinder inside another.
Formula: V = πh(R² – r²)
Alternative form: V = πh(R + r)(R – r)
Where:
- R = Outer radius (larger circle)
- r = Inner radius (smaller circle)
- h = Height
Real-life examples: Water pipes, steel tubes, circular bangles, cylindrical containers with walls.
Important: Always subtract the inner volume from the outer volume.
Volume Using Diameter
When the problem gives you diameter instead of radius, use these formulas:
Formula: V = π(d/2)²h
Simplified: V = πd²h/4
Where d = diameter
Why it works: Since radius = diameter/2, we substitute r = d/2 into the basic formula.
Quick tip: If diameter is given, divide it by 2 first to find radius, then use the standard formula.
Step-by-Step Examples
Example 1: Basic Volume
Find the volume of a cylinder with radius 7 cm and height 10 cm.
- Given: r = 7 cm, h = 10 cm
- Formula: V = πr²h
- Solution: V = (22/7) × 7² × 10
- V = (22/7) × 49 × 10
- V = 22 × 7 × 10 = 1,540 cm³
Example 2: Using Diameter
A cylindrical tank has diameter 14 m and height 5 m. Find its volume.
- Given: d = 14 m, h = 5 m
- First find radius: r = 14/2 = 7 m
- Formula: V = πr²h
- Solution: V = (22/7) × 7² × 5 = 770 m³
Example 3: Hollow Cylinder
A pipe has outer radius 5 cm, inner radius 3 cm, and length 20 cm. Find the volume of material.
- Given: R = 5 cm, r = 3 cm, h = 20 cm
- Formula: V = πh(R² – r²)
- Solution: V = (22/7) × 20 × (25 – 9)
- V = (22/7) × 20 × 16 = 1,005.71 cm³
Common Mistakes Students Make
- Mistake 1: Confusing radius and diameter always check whether the problem gives radius or diameter. Remember: radius = diameter ÷ 2.
- Mistake 2: Forgetting to square the radius the formula is πr²h, not πrh. The radius must be squared.
- Mistake 3: Wrong units in final answer volume is always in cubic units (cm³, m³, etc.), not square units.
- Mistake 4: Mixing up hollow and solid cylinder formulas For hollow cylinders, always subtract inner volume from outer volume.
- Mistake 5: Incorrect value of π Use π = 22/7 for calculations with multiples of 7, and π = 3.14 otherwise.
Quick Memory Tricks
- Trick 1: Circle × Height Volume = (Area of base circle) × Height. Remember: circle area first, then multiply by height.
- Trick 2: “R-squared, H-Paired” Radius gets squared (r²), height stays as is (h).
- Trick 3: Hollow = Outer – Inner For hollow cylinders, think “big minus small” for the radii.
- Trick 4: Diameter? Divide by 2 See diameter? Your first step is always d ÷ 2 = r.
FAQs on Volume of Cylinder
Q. What is the formula for finding volume of a cylinder?
The basic formula is V = πr²h, where r is the radius of the base and h is the height. This calculates the space inside the cylinder by multiplying the circular base area by the height.
Q. How do you find the volume of a cylinder using diameter?
Use the formula V = πd²h/4, where d is diameter and h is height. Alternatively, convert diameter to radius (r = d/2) and use the standard formula V = πr²h.
Q. What is the volume of hollow cylinder formula?
For hollow cylinders, use V = πh(R² – r²), where R is the outer radius, r is the inner radius, and h is the height. This gives the volume of the material forming the cylinder.
Q. Why do we square the radius in the cylinder volume formula?
We square the radius because the base is a circle, and the area of a circle is πr². Volume equals base area times height, so radius must be squared in the calculation.
Q. Can we calculate cylinder volume if only circumference is given?
Yes! Use the formula V = C²h/4π, where C is the base circumference and h is height. Or find radius first using r = C/2π, then apply the standard formula.
Q. What is the difference between lateral surface area and volume of a cylinder?
Lateral surface area (2πrh) measures the curved surface only in square units. Volume (πr²h) measures the space inside the entire cylinder in cubic units. They serve completely different purposes.
Q. How is the volume of a half cylinder calculated?
The volume of a half cylinder (semi-cylinder) is V = πr²h/2, which is exactly half of a full cylinder’s volume. Imagine cutting a cylinder vertically down the middle.
Q. What happens to cylinder volume if radius is doubled?
If radius doubles, volume increases by 4 times (not 2 times), because radius is squared in the formula. For example, if r becomes 2r, then r² becomes 4r².
Q. Is the formula for calculating volume of cylinder same for all types?
The basic formula V = πr²h applies to solid right circular cylinders. Hollow cylinders, oblique cylinders, or partial cylinders require modified formulas based on their specific geometry and given parameters.
Q. What units should I use for cylinder volume?
Always use cubic units for volume: cm³, m³, mm³, etc. Make sure all measurements (radius and height) are in the same unit before calculating, then cube that unit.
Conclusion
Mastering the volume of cylinder formula opens doors to solving countless real-world problems from calculating tank capacity to understanding engineering designs. Remember the core formula V = πr²h, and you’ll handle 90% of cylinder problems effortlessly.
Important notes:
- Always identify whether you have radius or diameter
- Square the radius before multiplying
- Use the hollow cylinder formula for pipes and tubes
- Keep units consistent throughout calculations
With these formulas in your toolkit and practice in your routine, cylinder problems become simple stepping stones to mathematical confidence. Whether it’s your homework, board exams, or competitive tests, you’re now fully equipped to tackle any cylinder volume question that comes your way.
