Finding the volume of a test tube or NMR tube is a common chemistry calculation, both in the lab for practical reasons and in the classroom to learn how to convert units and report significant figures. Here are three ways to find the volume.

## Calculate Density Using Volume of a Cylinder

A typical test tube has a rounded bottom, but NMR tubes and certain other test tubes have a flat bottom, so the volume contained in them is a cylinder. You can get a reasonably accurate measure of volume by measuring the internal diameter of the tube and the height of the liquid.

- The best way to measure the diameter of a test tube is to measure the widest distance between the inside glass or plastic surfaces. If you measure all the way from edge to edge, you'll include the test tube itself in your measurements, which isn't correct.
- Measure the volume of the sample from where it starts at the bottom of the tube to the base of the meniscus (for liquids) or the top layer of the sample. Don't measure the test tube from the bottom of the base to where it ends.

Use the formula for the volume of a cylinder to perform the calculation:

V = πr^{2}h

where V is volume, π is pi (about 3.14 or 3.14159), r is the radius of the cylinder and h is the height of the sample

The diameter (which you measured) is twice the radius (or radius is one-half diameter), so the equation may be rewritten:

V = π(1/2 d)^{2}h

where d is diameter

## Example Volume Calculation

Let's say you measure an NMR tube and find the diameter to be 18.1 mm and height to be 3.24 cm. Calculate the volume. Report your answer to the nearest 0.1 ml.

First, you'll want to convert the units so they're the same. Please use cm as your units, because a cubic centimeter is a milliliter! This will save you trouble when it comes time to report your volume.

There are 10 mm in 1 cm, so to convert 18.1 mm into cm:

diameter = (18.1 mm) x (1 cm/10 mm) [note how the mm cancels out]

diameter = 1.81 cm

Now, plug in the values into the volume equation:

V = π(1/2 d)^{2}h

V = (3.14)(1.81 cm/ 2)^{2}(3.12 cm)

V = 8.024 cm^{3} [from the calculator]

Because there is 1 ml in 1 cubic centimeter:

V = 8.024 ml

But, this is unrealistic precision, given your measurements. If you report the value to the nearest 0.1 ml, the answer is:

V = 8.0 ml

## Find the Volume of a Test Tube Using Density

If you know the composition of the contents of the test tube, you can look up its density to find the volume. Remember, density equal mass per unit volume.

Get the mass of the empty test tube.

Get the mass of the test tube plus the sample.

The mass of the sample is:

mass = (mass of filled test tube) – (mass of empty test tube)

Now, use the density of the sample to find its volume. Make sure the units of density are the same as those of the mass and volume you want to report. You may need to convert units.

density = (mass of sample) / (volume of sample)

Rearranging the equation:

Volume = Density x Mass

Expect error in this calculation from your mass measurements and from any difference between the reported density and the actual density. This usually happens if your sample isn't pure or the temperature is different from the one used for the density measurement.

## Finding the Volume of a Test Tube Using a Graduated Cylinder

Notice a normal test tube has a rounded bottom. This means using the formula for the volume of a cylinder will produce an error in your calculation. Also, it's tricky trying to measure the internal diameter of the tube. The best way to find the volume of the test tube is to transfer the liquid to a clean graduated cylinder to take a reading. Note there will be some error in this measurement, too. A small volume of liquid may be left behind in the test tube during transfer to the graduated cylinder. Almost certainly, some of the sample will remain in the graduated cylinder when you transfer it back to the test tube. Take this into account.

## Combining Formulas to Get Volume

Yet another method to get the volume of a rounded test tube is to combine the volume of a cylinder with half the volume of the sphere (the hemisphere that is the rounded bottom). Be aware that the thickness of the glass at the bottom of the tube may be different from that of the walls, so there is an inherent error in this calculation.