Measuring Density of Regular and Irregular Solids

To measure the density of a solid, find its mass with a balance and its volume either by calculation from measured dimensions (regular solids) or by the volume of water it displaces (irregular solids), then divide mass by volume: ρ = m ÷ V. In brief: for regular shapes use a ruler or caliper and a volume formula; for irregular solids that sink in water, use a measuring cylinder or an overflow (eureka) can. For solids that float, a sinker is added, or a denser liquid is used. Repeat the measurement and estimate the uncertainty.

Density can be used to identify materials and check purity. The method for measuring it is a standard school experiment that practices measurement of mass, length and volume and the analysis of uncertainty. This guide describes the procedures for regular and irregular solids, compares the methods in a table, works through an experiment with an uncertainty calculation and corrects common mistakes.

How do you measure the density of a regular solid?

  1. Measure the mass m on a balance (tared, on a flat surface).
  2. Measure the dimensions with a ruler (length, width, height of a cuboid) or a vernier caliper (for greater precision), and for a cylinder the diameter and height. Repeat at several places, since the sides may vary, and take the mean.
  3. Calculate the volume: V = lwh for a box, V = πr2h for a cylinder, V = (4/3)πr3 for a sphere.
  4. Calculate ρ = m ÷ V.

How do you measure the density of an irregular solid?

An irregular solid, such as a stone, cannot be measured by formula, so the displacement method is used:

  1. Measure the mass of the solid on a balance.
  2. Half-fill a measuring cylinder with water and read the volume V1 (at the bottom of the meniscus).
  3. Tie the solid with a thread and lower it gently into the water so that it is fully immersed and no water splashes out. Read the new volume V2.
  4. The volume of the solid is V = V2 − V1. Calculate ρ = m ÷ V.

For larger objects, an overflow (eureka) can is filled to the spout; the water pushed out by the object is collected in a cylinder, and its volume equals the volume of the object. Another method, based on Archimedes’ principle, finds the volume from the apparent loss of weight in water.

Measuring cylinder before and after a stone is lowered in 20 20 40 40 60 60 40 mL 47 mL add stone V = 47 − 40 = 7 cm³ before after
Displacement method for the stone in the worked example. The water level rises from 40 mL to 47 mL when the stone is fully immersed, so the stone has a volume of 7 cm³ (1 mL = 1 cm³). With a mass of 56.0 g, its density is 8.0 g/cm³.

What if the solid floats?

If the solid is less dense than water, such as wood or wax, a small heavy sinker can be attached: first measure the volume of the sinker alone, then the volume of the sinker and the solid together, and subtract. If the solid absorbs water, coating it with wax or using a different liquid in which it does not dissolve avoids errors. Alternatively, the solid can be measured in a denser liquid, like brine.

Methods compared

Methods for finding the volume of a solid
Method Used for Main source of error
Ruler and formulaRegular solidsReading of dimensions, uneven shape
Vernier caliperSmall regular solidsZero error
Displacement in a cylinderIrregular solidsMeniscus reading, air bubbles
Overflow canLarger irregular solidsWater lost or retained in the spout

Worked example: a stone and a metal cube

Part A: irregular stone. The mass of a stone is 56.0 g. Initial water level 40 mL, final level 47 mL (1 mL = 1 cm3), each read to ±0.5 mL.

  1. V = 47 − 40 = 7 cm3.
  2. ρ = 56.0 ÷ 7 = 8.0 g/cm3 = 8,000 kg/m3.
  3. Uncertainty: the volume has an uncertainty of ±0.5 ± 0.5 = ±1 mL, which is 1 ÷ 7 = 14 percent. The mass uncertainty (±0.1 g) is only 0.2 percent. The percentage uncertainty of the density is about 14 percent, so ρ = (8.0 ± 1.1) g/cm3.
  4. Improvement: use a larger stone, or a narrower cylinder with finer divisions, to reduce the percentage error in volume.

Part B: regular cube. A metal cube has sides measured with a caliper as 2.00 cm, and a mass of 62.4 g.

  1. V = 2.003 = 8.00 cm3.
  2. ρ = 62.4 ÷ 8.00 = 7.80 g/cm3, close to the density of steel (about 7.8 g/cm3).

How were these methods developed?

The displacement method is associated with Archimedes, in the legend of the golden crown (third century BCE). Overflow cans and measuring cylinders became standard in teaching laboratories in the 19th century. Modern laboratories use gas pycnometers and X-ray methods for porous solids.

Which mistakes are common?

Mistake 1: Letting water splash out

Lower the solid slowly with a thread.

Mistake 2: Leaving air bubbles on the solid

They add to the displaced volume and give a lower density. Tap the cylinder gently to remove them.

Mistake 3: Using too large a cylinder

A wide cylinder gives a small change in level, and a poor reading. Use the narrowest cylinder that fits.

Mistake 4: Reading the meniscus from above

Read at eye level, at the bottom of the curve.