# Burritos, an exercise in simple mathematics.

Ryan Schenk sent out an e-mail this weekend to document his mathematical approach at optimizing burrito filling.

Let’s take a look at his work:

End result?

Pi*((.816*dT)/(2*Pi))^2 * 2 * sqrt( (dT/2)^2 – (.816*dT/2)^2)

or roughly 0.031*dT^3
(where dT is the diameter of the Tortilla.)

It should be noted that this is the theoretical maximum. Experimental results are forthcoming.

Still, what does this mean in layman’s terms?

Basically, the optimal filling ratio is a square with dimensions in a ratio of ~8.1:5.7 laid out with the longer dimension NORMAL to the axis of wrapping (wrappation axis)

SAY WHAT?!?!

It’s true! I have been wrapping my burritos completely wrong! Readers, for heaven’s sake, don’t pack your burritos long, pack them wide! WIDE! This is the key to maximum filling.

Who knew?!

Math knew.

UPDATE: better scans and a third page of calculations
burrito 1.
burrito 2.
burrito 3.

### 14 thoughts on “Burritos, an exercise in simple mathematics.”

• 5/24/2011 at 10:00 am

I’m concerned that this method does not account for how much filling ends up in your lap when you try to eat a wide-filled burrito. A skinnier burrito may be the optimal method for maximizing how much filling is consumed.

• 5/24/2011 at 11:34 am

Jobonga, perhaps your burrito eating method could use improvement! Practice makes perfect you know!

• 5/24/2011 at 11:28 am

I have peer reviewed this work, and find that Schenk is correct!

I would like to add some refinement to the filling ratio, it exactly equals sqrt(2):1

• 5/24/2011 at 11:39 am

Visuals are necessary for the math challenged to understand all this.
Perhaps you could take the maximum filling, Ryan, and show how it works in a wide-filled rather than a skinny-filled burrito.
Also, would you let us know the diameter of the burrito and how much filling is the right (maximum) amount to put in? You may have addressed this on that sheet that is featured, but I don’t understand a word of it…

• 5/24/2011 at 11:56 am

Mom D,

I will be happy to oblige you with visuals. Perhaps tonight or tomorrow we shall have a burrito night to test this hypothesis in the real world!

• 5/24/2011 at 12:27 pm

Ryan, I just KNEW that you would rise to the challenge!

Looking forward to the photos –

• 5/24/2011 at 3:18 pm

Can I bring these blueprints into Chipotle and have them optimize my burrito?

• 5/24/2011 at 3:56 pm

good news!

The maximum volume solution is also a burrito that will close at both ends, there is a 63% safety factor as well.

• 5/24/2011 at 5:57 pm

Hmm… i’m not sure that i can approve of the peer reviewed calculations for maximum burrito filling if they’re only expressed in terms of area…

I think that the follow up post should indicate max volume, with dimensions, for the popular tortilla sizes: 6″, 8″ and 10″.

Only then, can we move on to the next phase, how do you pack the most flavor into that volume??

I’m thinking a strong foundation of grilled steak, followed by rice, cheese and a refried bean mortar to seal it in.
That said, the more granular shredded beef may allow a larger protein to filler ratio!
Who knows!

• 5/24/2011 at 9:50 pm

Jon,

If you follow the math, the optimal volume of stuffing is ~0.25*r^3, where r is the radius of the tortilla.

The guideline for layout that this provides is that the length of the burrito is about 15% longer than the radius of the tortilla.

• 5/25/2011 at 2:47 pm

This is great math, but terrible burrito. Like Jon Abad says, volume of the filling is COMPLETELY disgarded. It is a good first step though. Additions requested:

1. A simple volume of the burrito divided by the area of tortilla covered would give the thickness, and thus volume required. Easy addition.

2. The end flaps (previously disregarded), must have sufficient coverage to keep the filling theoretically within it, i.e. (dT-lB)/2 > dB ([the diameter of the tortilla minus the length of the burrito] divided by two must be greater than the diamater of the burrito)

• 5/25/2011 at 5:13 pm

Aaron,

1. The problem is seeking to maximize volume enclosed, as I replied to Jon’s comment that volume is ~0.25*r^3. The desired thickness would be ~0.13*r.

2. This treatment does disregard the end flaps being able to cover, however I checked the solution for coverage, and it will cover with 62% safety factor.