How this instrument works
Density is mass packed into space — how much stuff occupies a given volume. For a right circular cylinder, that volume is the circular cross-section times the height: V = πr²h. Fold that straight into the density ratio and the formula becomes ρ = m ⁄ (πr²h): weigh the piece, measure its radius and its height, and divide. Nothing about the shape's proportions matters beyond those two numbers — a short fat cylinder and a tall thin one with the same πr²h carry the same density if their masses match.
The radius is squared and the height is not, and that asymmetry has a practical consequence: a measurement slip on the radius costs twice as much, proportionally, as the same slip on the height. Read a 5.0 cm radius as 5.1 cm — a 2% error — and the computed volume, and hence the density, is off by roughly 4%. Calipers on the radius deserve more care than the tape on the height. This is the same reason a machinist chucking an unmarked bar of round stock measures diameter with a micrometer rather than a ruler before working out what alloy is sitting in the lathe.
The formula assumes a solid piece with no internal void. Feed it a length of pipe using the outer radius and it silently understates the metal's real density, because the volume term counts the hollow bore as if it were filled — the true fix is to subtract π(router² − rinner²)h before dividing by mass. That gap is a common source of confusion when someone weighs a length of tubing, plugs in the outer diameter, and concludes the metal is lighter than it actually is.
- Weigh the cylinder and enter the value in Mass, switching the unit menu to grams for small samples.
- Measure the radius — half the diameter, not the full width — with calipers and enter it in Cylinder radius.
- Measure the height along the cylinder's axis and enter it in Cylinder height.
- Read the result in Density; switch its unit menu to g/cm³ to compare directly against a materials table.
Worked example — an unmarked 4 kg bar of round stock
Set Mass to 4 kg, Cylinder radius to 5 cm, and Cylinder height to 10 cm — a squat cylinder about the footprint of a large coffee tin, but solid metal rather than a hollow can. Converting to metres first, radius is 0.05 m and height is 0.1 m, so the volume is π × 0.05² × 0.1 = 0.0007853981634 m³. Dividing the mass by that volume gives 4 ⁄ 0.0007853981634 = 5,092.95817894 kg/m³, the figure the calculator returns.
Switch the Density field to g/cm³ and the same reading becomes 5.093 g/cm³. That number rules out the light structural metals outright — aluminium sits at 2.70 g/cm³ and titanium at 4.51 g/cm³ — and lands among the denser mid-weight alloys, which is exactly the kind of narrowing-down a machinist wants before cutting into a bar that lost its mill-test paperwork somewhere along the way.
Questions
Why does radius appear squared in the density formula?
Because the cylinder's cross-sectional area is πr² — the squaring comes from area, not from density itself. The practical consequence is that a small error in measuring the radius has an outsized effect on the result: mismeasure the radius by 2% and the computed volume, and therefore the density, is off by roughly 4%, twice the relative error. Height enters only linearly, so calliper care spent on the radius pays off more than the same care spent on the height.
What happens if the cylinder is hollow, like a pipe?
The reading comes out too low. This formula assumes a solid disc filling the entire radius, so feeding it a pipe's outer radius divides the real mass by more volume than the metal actually occupies, understating the material's true density. To find the metal's own density from a tube, first work out the material's volume as π × height × (outer radius squared minus inner radius squared), then divide the mass by that smaller figure instead.
Can I enter diameter instead of radius?
Halve it first — the field wants the radius. Because r is squared in the formula, entering a diameter by mistake quadruples the apparent volume and understates density by a factor of four. A cylinder measured as 10 cm across has a 5 cm radius, and 5 cm is the number that belongs in Cylinder radius.
Why does the result default to kg/m³ instead of g/cm³?
kg/m³ is what falls straight out of feeding kilograms and metres into ρ = m ⁄ (πr²h) with no extra conversion factor, which is why it is the SI unit for density. g/cm³ is the friendlier scale for comparing against a materials table — water reads 1.00 g/cm³ either way — so the Density field's unit menu switches between the two; dividing a kg/m³ figure by 1,000 gives g/cm³ directly.
How is this different from just reading a mass off a scale?
A scale reports mass alone, and mass by itself does not identify a material. Two cylinders can both weigh 4 kg — a fat, short titanium blank and a slim, tall steel rod — while being made of entirely different metals; only dividing by volume separates them. Density is what lets samples of different sizes be compared on equal footing, which is why the calculator asks for radius and height as well as mass.
What if I enter a radius or height of zero?
The calculator refuses the input. A zero radius or height would put a zero in the denominator of ρ = m ⁄ (πr²h), an undefined division, so both dimensions must be entered as positive numbers before a density figure is returned.