Density is the amount of matter, or mass, per unit volume. This example problem shows how to calculate the mass of an object from a known density and volume.

## Simple Example (Metric Units)

As an example of a simple problem, find the mass of a piece of metal that has a volume of 1.25 m^{3} and a density of 3.2 kg/m^{3}.

First, you should notice both the volume and the density use the volume of cubic meters. That makes the calculation easy. If the two units were not the same, you'd need to convert one so they would be in agreement.

Next, rearrange the formula for density to solve for mass.

Density = Mass ÷ Volume

Multiply both sides of the equation by volume to get:

Density x Volume = Mass

or

Mass = Density x Volume

Now, plug in the numbers to solve the problem:

Mass = 3.2 kg/m^{3} x 1.25 m^{3}

If you see the units won't cancel out, then you know you did something wrong. If that happens, rearrange the terms until the problem works. In this example, cubic meters cancels out, leaving kilograms, which is a mass unit.

Mass = 4 kg

## Simple Example (English Units)

Find the mass of a blob of water with a volume of 3 gallons. It seems easy enough. Most people memorize the density of water as 1. But that's in grams per cubic centimeters. Fortunately, it's easy to look up the density of water in any unit.

Density of Water = 8.34 lb/gal

So, the problem becomes:

Mass = 8.34 lb/gal x 3 gal

Mass = 25 lb

## Problem

The density of gold is 19.3 grams per cubic centimeter. What is the mass of a bar of gold in kilograms that measures 6 inches x 4 inches x 2 inches?

## Solution

Density is equal to the mass divided by the volume.

D = m/V

where

D = density

m = mass

V = volume

We have the density and enough information to find the volume in the problem. All that remains is to find the mass. Multiply both sides of this equation by the volume, V and get:

m = DV

Now we need to find the volume of the gold bar. The density we have been given is in grams per cubic centimeter but the bar is measured in inches. First, we must convert the inch measurements to centimeters.

Use the conversion factor of 1 inch = 2.54 centimeters.

6 inches = 6 inches x 2.54 cm/1 inch = 15.24 cm.

4 inches = 4 inches x 2.54 cm/1 inch = 10.16 cm.

2 inches = 2 inches x 2.54 cm/1 inch = 5.08 cm.

Multiply all three of these numbers together to get the volume of the gold bar.

V = 15.24 cm x 10.16 cm x 5.08 cm

V = 786.58 cm^{3}

Place this into the formula above:

m = DV

m = 19.3 g/cm^{3} x 786.58 cm^{3}

m = 14833.59 grams

The answer we want is the mass of the gold bar in kilograms. There are 1000 grams in 1 kilogram, so:

mass in kg = mass in g x 1 kg/1000 g

mass in kg = 14833.59 g x 1 kg/1000 g

mass in kg = 14.83 kg.

## Answer

The mass of the gold bar in kilograms measuring 6 inches x 4 inches x 2 inches is 14.83 kilograms.

## Tips for Success

- The biggest problem students make when solving for mass is not setting up the equation correctly. Remember, mass equals density multiplied by volume. This way, the units for volume cancel out, leaving the units for mass.
- Be sure the units used for volume and density work together. In this example, the mixed metric and English units were intentionally used to show how to convert between units.
- Volume units, in particular, can be tricky. Remember, when you determine volume, you need to apply the correct formula.

## Summary of Density Formulas

Remember, you can arrange one formula to solve for mass, density, or volume. Here are the three equations to use:

- Mass = Density x Volume
- Density = Mass
**÷**Volume - Volume = Mass
**÷**Density

## Learn More

For more example problems, use the Worked Chemistry Problems. It contains over 100 different worked example problems useful for chemistry students.

- This density example problem shows how to calculate the density of a material when the mass and volume are known.
- This example problem shows how to find the density of an ideal gas when given the molecular mass, pressure, and temperature.
- This example problem shows the steps necessary to convert inches to centimeters.

## Source

- "CRC Handbook of Tables for Applied Engineering Science," 2nd Edition. CRC Press, 1976, Boca Raton, Fla.