How this instrument works
A cylinder's volume follows from stacking: take the circular base, area πr², and multiply by however many units of height are piled on top of it. That is the entire content of V = πr²h — an instance of Cavalieri's principle, the seventeenth-century result stating that if every cross-section of one solid matches the area of the corresponding cross-section of another, the two solids share the same volume. A cylinder is simply a right prism whose repeating cross-section happens to be a circle instead of a triangle or square.
The liter itself is not a separate system of measurement bolted onto the metric one; it is a special name for a single cubic decimeter, fixed by the General Conference on Weights and Measures in 1964. Because one cubic decimeter equals exactly 1,000 cubic centimeters, converting a raw πr²h answer into liters means multiplying a cubic-meter result by 1,000, or dividing a cubic-centimeter result by the same number — the arithmetic this sheet performs automatically so the figure printed matches what's stamped on a bottle label.
Radius and height do not pull equal weight, and that asymmetry catches people out: doubling the Height only doubles the volume, while doubling the Radius quadruples it, because r is squared and h is not. Push the radius toward zero and the cylinder collapses into a line segment with no cross-section left to hold anything, so the volume falls to exactly zero no matter how tall the shape still stands — the formula enforces that limit without needing a special case bolted on.
- Enter the cylinder's Radius — the distance from the center of the circular base to its rim, not the width measured straight across.
- Enter the Height — the perpendicular distance between the two circular faces, measured straight up the axis.
- Switch the unit shown beside Radius or Height to mm, m, or in if your tape measure reads in a different unit; the sheet converts before it multiplies.
- Read Volume for the result, reported in liters by default — flip its unit to milliliters or gallons if that suits the container in front of you.
Worked example — a 5 cm radius, 10 cm tall cylinder
Picture a squat measuring cup with Radius set to 5 centimeters (0.05 m) and Height set to 10 centimeters (0.1 m). The formula multiplies base area by height: V = π × 0.05² × 0.1 = 0.0007853981633974484 cubic meters — a correct but unhelpful figure until it's rescaled into a unit anyone reads on a label.
Multiplying by 1000 converts cubic meters to liters, since a liter is defined as one-thousandth of a cubic meter: 0.0007853981633974484 × 1000 = 0.7853981633974484 liters, which rounds to about 0.785 liters. That is a bit under a standard soda can's contents, and it is exactly what this sheet reports as Volume for those two inputs.
Questions
What is the formula for a cylinder's volume?
V = πr²h, where r is the radius of the circular base and h is the perpendicular height. It is simply the base's area, πr², repeated h times — the same base-times-height logic used for any prism, just applied to a circular cross-section instead of a polygon.
Why does the answer come out in liters instead of cubic centimeters?
Because a liter is officially defined as one cubic decimeter — a 10 cm cube — which makes it 1000 cubic centimeters exactly. This sheet computes πr²h in whatever unit you entered, then rescales the raw cubic figure into liters automatically, since liters are what a can, bottle, or tank label actually uses.
Does doubling the radius change the volume as much as doubling the height?
No, and this is the mistake that trips people up most often. Height enters the formula to the first power, so doubling it doubles the volume. Radius is squared, so doubling it quadruples the volume. A wide, short cylinder can hold far more than a narrow, tall one built from the same amount of material.
What happens to the volume if the radius is set to zero?
The volume drops to exactly zero, regardless of how large the height is. A zero-radius cylinder has no circular cross-section at all — geometrically it has collapsed into a straight line segment — so there is no base area left for the height to multiply.
How is a cylinder's volume different from a general prism's?
It isn't, structurally — a cylinder is a right prism whose repeating cross-section is a circle rather than a triangle or rectangle. Both follow Cavalieri's principle: stack a constant cross-sectional area up through the height and the running total is the volume, whatever shape that cross-section takes.
Which unit should I enter Radius and Height in?
Any consistent length unit works, since the sheet converts internally before multiplying — millimeters, centimeters, meters, or inches all give the correct Volume once converted to liters. Just avoid mixing a radius read off a ruler in inches with a height read off a different tool in centimeters.