# How does the size of a cookie depend on the size of the ball of dough?

This term I’m teaching Calculus 3 which involves learning about the concept of curvature. This is a measurement of how bendy or curvy something is. The flatter something is, the less curvature it has.

We learn in class that a circle or sphere of radius r has curvature inversely proportional to its radius, that is it has curvature $\frac{1}{r}$.

In this class we used baking cookies to illustrate how the curvature of an object can change over time. Seen from over top, a ball of cookie dough flattens out as it bakes.

This got me thinking about how exactly is the size of the ball of cookie dough related to the size of the cookie you get in the end? So I did some science.

## The recipe

A student in my class provided me with the following recipe:

Here’s a simple peanut butter recipe that’s safe for people with gluten and/or dairy allergies:

If change in curvature is desired:

• 1 cup peanut butter
• 3/4 – 1 cup sugar
• 1 egg
• 1 tsp baking soda
• tiny splash of vinegar
• tiny pinch of salt

Preheat oven to 350 degrees. Roll dough into balls and place on cookie sheet. Bake for ~10-12 minutes.

For those who desire nearly constant curvature:

• 1 cup peanut butter
• 1 cup sugar
• 1 egg
• pinch of salt

Preheat oven to 350 degrees. Roll dough into balls, place on cookie sheet and flatten to desired curvature with fork. Bake for ~10-12 minutes.

After mixing everything together (using 3/4 cup sugar), since I used natural peanut butter the dough was too goopy to form into balls. So I made the following additions:

• 1/2 cut oats
• 1/4 cup cricket powder (for protein)
• 1 tsp cinnamon

## Baking and data

For my first batch I planned to take them out after 11 minutes, but they needed additional time, so I left them in the oven for an additional 5 minutes. This could potentially introduce some p-value hacking because I changed my experiment in the middle of it. I don’t think the additional time changed the shape of the cookies, just how gooey they were in the inside.

I got the following results for Batch 1:

 Diameter of dough ball (cm) Diameter of cookie (cm) 2 4.5 2 4.5 2.5 5 3 6 3 6.5 3 6 3.5 6.5 3.5 7 4 8 3.5 8 4 8 4.5 9.5 5.5 11

The three biggest cookies pushed into each other and didn’t spread out completely. This made them a little more square than they should have been.

Batch 2 was in for a full 16 minutes, but it needed even more time! I put them in for an additional 4 minutes.

Here are those results:

 Diameter of dough ball (cm) Diameter of cookie (cm) 5.5 11 6.5 14.5

The biggest cookie was pretty unstable at first, but after leaving it on the pan a little longer it firmed up.

## Data analysis and results

Here’s what the cookies look like stacked from largest diameter to smallest.

Of course I had to plot this data, so I did and got the following line of best fit:

In English:

The diameter of a cookie is twice the diameter of the ball of dough used to make it.

In terms of radius, since the radius is half the diameter, and they compound, we get:

The radius of the cookie is four times the radius of the ball of dough.

Since curvature is inversely proportional to curvature, we get:

The curvature of the cookie is a quarter the curvature of the ball of dough.

## Conclusions

I think we can actually make some interesting conclusions about this.

What size cookies should I make to avoid wasted space on my cookie sheet?

It turns out that by using this relationship between the size of the dough ball and the size of the cookie, if you have a fixed amount of dough V, and a fixed area of your cookie sheet you should make:

$n=\frac{\pi^2 A^3}{2304 V^2 }$ cookies of diameter $d = \frac{24V}{\pi A}$.

I hope you had as much fun as I did! Thanks for reading.

Thanks to Robert Fajber for improvements to the graph, and Jessie Lamontagne for further directions and questions.

## Math Appendix

If you’re interested in the nitty-gritty details about how I came up with those formulas, here they are.

Fix $A,V$. Assume we want n cookies of diameter D ( that start with diameter d). We know $2d = D$. We want to space out the cookies so that their bounding squares do not overlap. These squares give us

$A = nD^2$

The volume of the dough gives us

$V = n \frac{4\pi}{3} (\frac{d}{2})^3 = n \frac{4\pi}{3}(\frac{D}{4})^3 = n \frac{\pi}{48}D^3$

This is two equations and two unknowns. Solving that gives us the desired formulas for D and n. Then we related D back to d.

## One thought on “How does the size of a cookie depend on the size of the ball of dough?”

1. A good way to falsify your theory would be to measure the thickness of the cookies.
Infact given a volume of dough $V$ (for a single cookie), and calling $C$ a baking-dependent constant which relates the raw volume $V$ with the baked volume $V_1 = CV$ (so we expect $C <1$, because of drying), then if $V_2$ is the volume of the final (approximately disc-shaped) cookie, we get

$V_1 = \frac{4\pi R_1^3}{3C} = \pi TR_2^2 = V_2$.

Using the empirical law $R_2 = 4R_1$, this simplifies to

$T = \frac{R_1^2}{12C}$

so thickness should be proportional to the square of the dough radius.

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