Tube Calculator: Hollow Cylinder Volume & Area

A hollow cylinder, built from two shapes you already know

A tube — also called a hollow cylinder or pipe — is a cylinder with a smaller cylindrical hole bored straight through its center. Slice straight across it and the cross-section you see is an annulus: a ring between two concentric circles, outer radius $r_1$ and inner radius $r_2$. Given $r_1$, $r_2$, and the height $h$, every other measure follows:

$t = r_1 - r_2$ $A = \pi(r_1^2 - r_2^2) \quad \text{(cross-section — the annulus)}$ $\boxed{V = \pi(r_1^2 - r_2^2)h = A \times h}$ $C_1 = 2\pi r_1 \qquad C_2 = 2\pi r_2$ $L_1 = 2\pi r_1 h \qquad L_2 = 2\pi r_2 h$

The volume is the outer cylinder's volume $\pi r_1^2 h$ minus the inner (hollow) cylinder's volume $\pi r_2^2 h$ — factor out $\pi h$ and you get exactly $\pi(r_1^2-r_2^2)h$, the annulus cross-section times the height.

Worked example: r1 = 5, r2 = 3, h = 10

$t = 5 - 3 = 2$ $A = \pi(25 - 9) = 16\pi \approx 50.2655$ $V = \pi(25-9)(10) = 160\pi \approx 502.6548$ $C_1 = 2\pi(5) \approx 31.4159 \qquad C_2 = 2\pi(3) \approx 18.8496$

The wall is 2 units thick, the ring-shaped cross-section is exactly $16\pi$, and multiplying by the height $10$ gives a volume of exactly $160\pi \approx 502.65$ cubic units.

Common mistakes

• Using $(r_1-r_2)^2$ for the cross-section area. The correct form squares each radius FIRST, then subtracts: $r_1^2 - r_2^2$, not $(r_1-r_2)^2$ — the same guard that applies to the annulus shape this cross-section actually is.
• Forgetting to subtract the hollow core. Computing the volume as a solid cylinder $\pi r_1^2 h$ and skipping the inner subtraction overstates the volume — a tube is never as heavy (or as full) as a solid rod of the same outer radius.
• Confusing the wall thickness $t$ with a radius. $t = r_1 - r_2$ is a length, not a radius — it's the distance across the solid wall, not a distance from the center.

Where this shows up

• Water pipes, steel tubing, pipe insulation: volume tells you how much material (metal, plastic, foam) makes up the wall — useful for weight, cost, or material estimates.
• Cardboard tube cores, washers, spacers: any hollow cylindrical part where the inner hole matters as much as the outer size.
• Sanity-checking against a plain cylinder: as $r_2 \to 0$, the tube's formulas reduce exactly to the cylinder's own $V = \pi r^2 h$ — a useful check that the tube math is set up correctly.

A tube is a cylinder hollowed out along its axis — knowing the outer radius, inner radius, and height is always enough to determine the wall thickness, cross-section area, volume, and both circumferences, with the annulus cross-section carrying the same difference-of-squares guard as its standalone counterpart.