Volume of Cylinders: Formula, Examples & Hollow Cases

A cylinder is a can-shaped solid with two circular ends and a straight side — think of a water pipe, an oil drum or a battery. Its volume is the area of the circular base multiplied by the height: every thin circular slice of the cylinder has area πr², and stacking h units of them gives πr²h. A hollow cylinder, like a pipe or a straw, is the same idea with the inner empty part subtracted.

This formula is the workhorse of capacity problems. Oil barrels, water pipes, test tubes, syringes and storage tanks are all cylinders. Hollow cylinders add one more practical skill: pipes are sized by their outer and inner radii, and the material in the wall of the pipe is the difference between the outer and inner volumes.

Key Concepts

• Radius (r): the radius of the circular base.
• Height (h): the distance between the two circular ends.
• Base area: πr² — every formula for cylinder volume starts here.
• Hollow cylinder: a tube with outer radius R and inner radius r; its wall material is the volume between the two cylinders.
• Volume is measured in cubic units (cm³, m³), and 1000 cm³ = 1 litre.

Formulas

• Volume of a solid cylinder: V = πr²h — base area times height.
• Volume of material in a hollow cylinder: V = π(R² − r²)h — the outer cylinder minus the inner empty cylinder.
• Capacity of a hollow cylinder: V = πr²h, using the inner radius r, gives the space the pipe can actually hold.
• These assume a right circular cylinder with the axis perpendicular to the base.

Worked Example

Find the volume of a cylinder with radius 7 cm and height 10 cm. Use π = 22 ÷ 7.

1. Write the formula: V = πr²h = (22 ÷ 7) × 7² × 10.
2. Compute 7² = 49, and (22 ÷ 7) × 49 = 22 × 7 = 154.
3. Multiply by the height: 154 × 10 = 1540. The volume is 1540 cm³, which is 1.54 litres.
4. Bonus — a hollow pipe with outer radius 10 cm, inner radius 8 cm and length 50 cm: wall volume = (22 ÷ 7) × (100 − 64) × 50 = (22 ÷ 7) × 36 × 50 = (22 ÷ 7) × 1800 = 39600 ÷ 7 ≈ 5657 cm³.
5. Its water capacity = (22 ÷ 7) × 64 × 50 = 70400 ÷ 7 ≈ 10057 cm³, or about 10.06 litres.

Common Misconceptions

• Forgetting to square the radius: the formula is πr²h, not πrh. Skipping the square gives a surface-area-like number, not a volume.
• Using the outer radius for capacity: the water inside a pipe is limited by the inner radius, not the outer one.
• Subtracting radii instead of their squares: the wall volume needs R² − r², not (R − r)²; the two are different numbers.

Real-World Applications

• Calculating the capacity of water tanks, oil drums and cooking-gas cylinders.
• Designing pipes: wall material from π(R² − r²)h, flow capacity from πr²h.
• Dosing in syringes, test tubes and graduated cylinders in laboratories.
• Estimating concrete for cylindrical pillars and wells.

A solid cylinder holds πr²h, and a hollow pipe's wall contains π(R² − r²)h while its capacity uses only the inner radius. Square the radius before multiplying, keep the units cubic, and convert to litres with 1000 cm³ = 1 litre.