Volume of a hollow cylinder
The volume of a hollow cylinder is V = π(R² − r²)h, where R is the outer radius, r the inner radius and h the length. You take the volume of the full outer cylinder and subtract the hole. A tube with R = 4 in, r = 3 in and h = 10 in has π × (16 − 9) × 10 ≈ 219.91 in³ of material in its wall.
Contents
V = π × (R² − r²) × h. With R = 4 in, r = 3 in and h = 10 in: π × 7 × 10 = 219.91 in³ of wall material. The bore holds π × 3² × 10 = 282.74 in³.
The calculator above accepts outer and inner radii, outer and inner diameters, or an outer diameter with a wall thickness. It shows the material volume, the volume of the bore and how much the bore holds.
What is a hollow cylinder?
A hollow cylinder is a cylinder with a smaller, coaxial cylinder removed from its center. Pipes, tubes, drinking straws, bushings, washers, concrete culverts and the cardboard core of a paper roll are all hollow cylinders. Its cross-section is not a disk but a ring, called an annulus.
Two volumes matter, and people mix them up all the time:
- Material volume (the wall): π(R² − r²)h. Use it for weight, cost of metal, plastic or concrete.
- Capacity (the bore): πr²h. Use it for how much water, air or grain the tube holds. That is an ordinary cylinder volume with the inner radius.
Properties of a hollow cylinder
| Quantity | Formula | R = 4, r = 3, h = 10 (in) |
|---|---|---|
| Wall thickness | t = R − r | 1 in |
| Ring (base) area | π(R² − r²) | 21.99 in² |
| Material volume | π(R² − r²)h | 219.91 in³ |
| Bore volume (capacity) | πr²h | 282.74 in³ |
| Outer cylinder volume | πR²h | 502.65 in³ |
| Outer curved surface | 2πRh | 251.33 in² |
| Inner curved surface | 2πrh | 188.50 in² |
| Two ring ends | 2π(R² − r²) | 43.98 in² |
| Total surface area | 2π(R + r)h + 2π(R² − r²) | 483.81 in² |
Check the first lines: material plus bore equals the outer cylinder, 219.91 + 282.74 = 502.65 in³. That simple sum is a good way to catch a typing error.
Derivation of the volume of a hollow cylinder
Every slice of a hollow cylinder cut square to its axis is the same ring. The ring is a big circle of area πR² minus a small circle of area πr², so its area is π(R² − r²). Stack that ring over the height h and you get V = π(R² − r²)h. This is the same "base area × height" rule that gives πr²h for a solid cylinder; the derivation of the cylinder volume formula covers it with disks and with an integral.
A second form is useful for thin walls. Factor the difference of squares: R² − r² = (R + r)(R − r). With the mean radius m = (R + r) ÷ 2 and the thickness t = R − r, the formula becomes V = 2π × m × t × h. This is exact, not an approximation. It says the wall is a flat sheet as long as the mean circumference 2πm, as wide as h and as thick as t, rolled into a tube.
How to calculate the volume of a hollow cylinder
- Get both radiiFrom diameters, halve them: R = D ÷ 2, r = d ÷ 2. From an outer diameter and a wall thickness, r = D ÷ 2 − t.
- Use one unitConvert the length to the same unit as the radii (feet to inches, meters to centimeters).
- Square and subtractCompute R² − r². Never square the thickness alone: (R − r)² is a different number.
- Multiply by π and hThe result is in cubic units of your measurement.
Example 1: steel tube in inches
A steel tube is 4 in across the outside, has a 0.25 in wall and is 10 ft long. R = 2 in, r = 2 − 0.25 = 1.75 in, h = 10 × 12 = 120 in.
R² − r² = 4 − 3.0625 = 0.9375 in². V = π × 0.9375 × 120 ≈ 353.43 in³, or 353.43 ÷ 1,728 ≈ 0.2045 ft³ of steel. Carbon steel weighs roughly 0.2836 lb per cubic inch (7.85 g/cm³), so the tube weighs about 100 lb. The cylinder weight calculator does this step for other materials.
The bore of the same tube holds π × 1.75² × 120 ≈ 1,154.5 in³, which is 1,154.5 ÷ 231 ≈ 5.00 US gallons.
Example 2: concrete pipe in centimeters
A concrete pipe section measures 60 cm outside, 50 cm inside and 2 m long. R = 30 cm, r = 25 cm, h = 200 cm.
V = π × (900 − 625) × 200 = π × 275 × 200 ≈ 172,788 cm³ ≈ 0.1728 m³ of concrete, or 172.8 L. The bore holds π × 25² × 200 ≈ 392,699 cm³, about 392.7 L of water.
Example 3: with diameters and the mean radius
D = 10 in, d = 6 in, h = 12 in. Direct: π × (25 − 9) × 12 ≈ 603.19 in³. With the mean radius m = 4 in and t = 2 in: 2π × 4 × 2 × 12 ≈ 603.19 in³. Both forms agree.
Volume of common tube sizes
Volumes for a 12 in length, outside diameter and wall thickness as listed. Values are computed, not taken from a product catalog.
| Outer diameter | Wall | Material (in³ per ft) | Bore (in³ per ft) | Bore (US gal per ft) |
|---|---|---|---|---|
| 1 in | 0.125 in | 4.12 | 5.30 | 0.0229 |
| 2 in | 0.25 in | 16.49 | 21.21 | 0.0918 |
| 4 in | 0.25 in | 35.34 | 115.45 | 0.4998 |
| 6 in | 0.5 in | 103.67 | 235.62 | 1.0200 |
For the water held by long pipe runs, the pipe volume calculator works directly in gallons per foot from the inside diameter.
Finding the thickness or the inner radius
If you know the volume of material, the outer radius and the length, rearrange: r = √(R² − V ÷ (πh)), then t = R − r. Example: R = 2 in, h = 12 in, V = 50 in³. V ÷ (πh) = 50 ÷ 37.699 ≈ 1.3263, so r = √(4 − 1.3263) ≈ 1.635 in and the wall is about 0.365 in thick. If V ÷ (πh) comes out larger than R², the numbers are inconsistent: no tube of that outside size can hold that much material.
Common mistakes
- Subtracting the radii before squaring. π(R − r)²h is wrong. For R = 4 and r = 3 it gives 31.4 in³ instead of 219.9 in³.
- Using diameters in the radius formula. The answer is four times too large. Use π(D² − d²)h ÷ 4 with diameters.
- Mixing inch walls and foot lengths. Convert first.
- Answering the wrong question. The wall volume tells you weight; the bore tells you capacity. A thick-walled tube can have far more material than space inside.
Quick check: the wall volume is always less than the outer cylinder πR²h, and wall + bore must add up to it exactly.
Asymmetric and partial hollow cylinders
If the hole is off center, the volume does not change: it is still the outer cylinder minus the hole, π(R² − r²)h, as long as the hole stays fully inside. Only the wall thickness varies around the tube. A half tube, like a split sleeve, is half that volume; the same angle rule as for a half cylinder applies: multiply by θ ÷ 360°. To check any single cylinder you compute along the way, use the cylinder volume calculator.
FAQ
What is the formula for the volume of a hollow cylinder?
V = π(R² − r²)h, where R is the outer radius, r the inner radius and h the length or height. It is the volume of the outer cylinder minus the volume of the hole. With diameters, V = π(D² − d²)h ÷ 4.
Does a hollow cylinder have two radii?
Yes. The outer radius R goes to the outside surface and the inner radius r goes to the inside surface of the bore. Their difference, R − r, is the wall thickness. Both are measured from the same center.
How do I find the volume of a hollow cylinder in liters?
Measure R, r and h in centimeters, compute π(R² − r²)h to get cubic centimeters, then divide by 1,000. A tube 60 cm across the outside, 50 cm across the inside and 200 cm long has 172,788 cm³ of wall, which is 172.8 L.
How do I calculate the wall thickness of a hollow cylinder from its volume?
Solve the formula for the inner radius: r = √(R² − V ÷ (πh)). The thickness is then t = R − r. For R = 2 in, h = 12 in and V = 50 in³, r ≈ 1.635 in and the wall is about 0.365 in thick.
How does the volume of a hollow cylinder change when the height is doubled?
It doubles. The height appears once in V = π(R² − r²)h, so the volume is directly proportional to it. Doubling both radii instead multiplies the volume by four.
What is the base area of a hollow cylinder?
The base is a ring (an annulus) with area π(R² − r²). For R = 4 in and r = 3 in, the ring covers π × 7 ≈ 21.99 in². Multiply it by the height to get the volume.
Related
- How to find the volumeHow to find the volume of a cylinder: square the radius, multiply by π and the height. Worked examples in inches, feet, cm and m, plus common mistakes.
- Pipe volumePipe volume calculator: V = π × r² × L from the inside diameter. Gallons of water per foot of pipe, liters per meter, a full chart and worked examples.
- Cylinder weightPipe weight calculator for round bars, tubes and cylinders: weight = volume × density. Steel, aluminum, copper, water and concrete, with worked examples.
Updated October 12, 2026