Our newest chickens are laying eggs but they look small. The first egg was so small that we ignored it. The next few eggs were wider but they were also shorter. Were the eggs getting bigger or smaller? — we couldn’t tell.

We’re not ungrateful for small eggs – we’re happy to have any at all. Since our Cuckoo Marans stopped laying their deep brown eggs over a year ago, we’ve been missing fresh eggs. A small green egg feels like a great gift to us.

But we were curious. Were these eggs getting any larger by the day? Height and diameter are pretty easy to measure so we started there.

## Measurements on the “Size”of an Egg

We measured the height and diameter with an electronic caliper, taking care to place the caliper across the largest section. The graph below shows how the height and diameter varied over time. You can hover over any point to see the measurement.[cf]js0[/cf][cf]js1[/cf][chartcaption caption=”Egg height and diameter as measured with an electronic caliper.” align=”aligncenter” width=”300″][/chartcaption]

Neither height nor length increased consistently. And, ignoring the first two eggs, I would have said the eggs were getting smaller.

So what’s more important, the length or the diameter? Is there something else that would express the “size” more clearly? It’s hard to think of both height and diameter at the same time. We needed a single measure that captures the effect of both measures into one.

## Measurements on the Volume of an Egg

The volume is a measure that includes the effect of both length and diameter. We can get the volume of an egg by placing it in a measuring cup then capturing the rise of water caused by fully submerging the egg. But water compromises the storage life of eggs. The water dissolves the egg’s protective coating. It is the coating that seals the hard but slightly porous shell and gives fresh eggs a non-refrigerated shelf life of weeks! Because I have breakfast plans for these eggs, we need another way to figure out the overall “size”.

## Measurements on the Weight of an Egg

Next, we tried to weigh the eggs because both the length and width ought to contribute to the weight. But an egg weighs only some tens of grams and all I had on-hand was a 20 kg scale. Would it be precise enough to tell the eggs apart?

[cf]js3[/cf][chartcaption caption=”Weight of the eggs as measured by an electronic scale.” align=”aligncenter” width=”300″][/chartcaption]

Although the scale is relativity precise for it’s capacity (20kg), it only discriminates in two gram increments. That’s not enough resolution to tell most of the eggs apart. But it is enough to determine that the egg size isn’t shrinking. This wasn’t obvious using the height or diameter measurement alone.

Still, unhappy with the resolution of the scale, and knowing that our length measurements are very precise, we decided to give geometry a try.

## The Geometry of an Egg

The egg is shaped a lot like a prolate spheroid (even better). That’s a fancy way of saying it’s like a squished sphere with a height that exceeds the diameter.

The volume of a prolate spheroid increases in proportion to any increase in length (we’ll call the length “h”). We’ll call the diameter “d”. The volume of the prolate spheroid increases even faster with diameter. This is a lot like pizza. I remember my friend Stephen telling me how as a kid he discovered that pizza’s value is determined by the area rather than the diameter. He was right and this applies to eggs as well.

The relationship of an eggs volume to its diameter (‘d’) and its height (‘h’) is expressed in the equation you see below. The volume is roughly equivalent to the height multiplied by the square of the diameter. However someone has magically dropped pesky constant values into the equation. It is not necessary to consider these constants when comparing eggs. If you are curious about the constants — they are related to the circular nature of the egg.

Calculating the volume by using the measured height and diameter and plotting it allows us to see how the volume of the egg varies over a ten day period. These values are precise and show that the last egg in the series is larger than all of the others.[cf]js2[/cf][chartcaption caption=”Egg volume calculated using the prolate spheroid formula.” align=”aligncenter” width=”300″][/chartcaption]

So hopefully you found these connections interesting. It was not a simple path to get the answer of which egg is largest, because they all look similar in size. Plus the size differences have to be measured in two dimensions (‘d’ and ‘h’) instead of one. This forced us to think critically about the common notion of size. With the help of math and measurement tools we are now certain about which eggs are bigger or smaller, and by how much.

Robert Goldhagen says

Why are you not measuring end to end of the shell?

andy says

Oh, I think I see what’s going on. I did measure end-to-end, but in setting down the caliper to take the picture, parallax error crept in and made it appear otherwise. Sorry for the misleading photo.