How to Calculate Density

What is density? How can you calculate it? What are some densities of common everyday stuff like water and air? And how can you use all of this new knowledge to understand what will and will not float in water? Those are exactly the questions we’ll be answering today.

What Is Density?

First things first: What is density? When we talk about the density of a certain material, we’re talking about how compactly the matter it’s made of is packed. Put more precisely and more mathematically, the density of a material is the amount of mass contained in a given volume—say the number of grams per cubic centimeter or kilograms per cubic meter.

Density gives us a consistent way to calculate and compare the compactness of different materials.

The mass of something is really just a measure of how much stuff that something contains. By “stuff” I mean the number of protons and neutrons contained in its atoms (since that’s where almost all the mass of an atom is). So the density of a material is a measure of how many protons and neutrons it contains in a given volume. High density materials like metals have lots of “stuff” per unit volume, while low density materials like the air we breathe have a lot less “stuff” in that same volume.

The important thing to know is that density is useful because it gives us a consistent way to calculate and compare the compactness of different materials.

Some Good Densities to Know

As a quick aside, it’s useful in life to know a few numbers by heart. One of those numbers that’s good to know is the density of water. Why? Mainly because it’s a convenient baseline material that we’re all familiar with, and as such it’s convenient to use as a comparison for densities of other materials.

Water has a density of 1 gram per cubic centimeter. This means that if you were holding a 1 cubic centimeter cube of water in your hand (that’s about the size of a sugar cube), it’d weigh 1 gram. This, as an added bonus, now gives you a good way to think about how much 1 gram weighs.

Just for comparison:

• The density of air at sea level is about 0.001 grams per cubic centimeter or about 1,000 times less dense than water
• The density of normal everyday rock is a little less than 3 grams per cubic centimeter or about 3 times the density of water
• The density of iron is just shy of 8 grams per cubic center or about 8 times the density of water

Just to totally blow your mind, the density of the hydrogen and helium gas at the center of the Sun is upwards of 150 grams per cubic centimeter. That’s about 150 times the density of water or 20 times the density of iron … and yet, because it is so hot, this material is still a gas!

How to Calculate Density

With this basic understand of density under our belt, let’s now move to the question of how you actually go about calculating the density of some material in the real world. If you understand the definition of density, you should pretty much already know the answer. Since the density of a given material is the amount of mass contained in a given volume of that material, it must be that you can calculate the density of some material by dividing its mass by its volume.

Calculate the density of some material by dividing its mass by its volume.

So, if you have a cube of something that measures 1 centimeter on each side, you can find its density by dividing its mass (in units of grams or kilograms) by its 1 cm x 1 cm x 1 cm = 1 cubic centimeter volume (commonly abbreviated 1 cc). If you have a sphere of material with a radius r, you just have to divide its mass by (4/3)πr³ (that’s the volume of a sphere) to find its density.

Of course, most real materials don’t come as cubes or spheres, which means calculating density is usually a little tougher in the real world. The tricky part about measuring the density of these real world objects is measuring their volume. One way to measure the volume of something is to place it into a tub of water and watch how much the water rises. This rise is directly related to the increased volume of the stuff in the tub, and the change in volume you measure must therefore equal the volume of the object you put into it. And thus, if you know the mass of the object, you can use this calculated volume to find the material’s density.

Why Do Humans Float?

With all of these practical considerations out of the way, let’s turn to the all-important question for today: How can you tell if something you put into a bathtub full of water will float? The answer turns out to be pretty simple. Any object whose density is less than the density of water will float in a pool of water, while any object whose density is greater than water will sink.

As we’ve seen, the density of fresh water is 1 gram per cubic centimeter (the density of salt water is a little higher), while the density of metals is many times greater than the density of water. How then is it possible that a huge ship that’s made out of dense metal can float? Because most of the ship isn’t made of metal—it’s made out of the stuff contained between the metal hull … namely air! The density of air is very low, so when you calculate the density of the entire ship, you’ll find that its density is lower than water. And thus a boat will float.

On the same note, this explains why people can also float in a pool of water. To begin with, humans are mostly made of water, which means that our densities must be right around that of water. But humans also have fairly sizable lungs that fill up with very low density air, which means that our densities are actually a bit less than water. And thus we too can float!

Wrap Up

OK, that’s all the math we have time for today.

For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Water image from Shutterstock.