The math of a circular sukkah

When I first studied Sukkot 8a in my studies at Midreshet Lindenbaum, I was warned it was a tough Gemara to crack. Without Artscroll to illuminate it, I struggled through it the first time. The mishna in Sukkah has lots of geometry, but most of it deals with Sukkot that are rectangular prisms. This Gemara asks, what if your Sukkah is circular? What will be its minimum size?

Many years later, I wound up with a new curriculum in my 6th grade class, who needed substantial enrichment. This curriculum happened to teach circumference in the third week of school - usually right before Sukkot. I went back to the Gemara in Sukkot 8a and asked, "Can I take this apart so that talented 6th graders can grasp its intricacies?"

The short answer is "somewhat." I designed an investigation that focuses on circumference, pi and unit conversions, while only delicately touching on the aspects of the Gemara that require an understanding of the Pythagorean Theorem. My worksheet begins where the Gemara does - with Rabbi Yochanan's assertion that a circular Sukkah needs to have enough space in its circumference to seat 24 people. (You can find a blank worksheet here and a partially completed student copy here.)

This lesson is really a two-day lesson, unless you have at least an hour for math. The first day is really active. I never have 24 students in my class, Baruch Hashem, so I usually either have the students make a circle of 24 chairs, or have 12 students sit in a circle and tell them to double all the numbers. The goal here is to get the kids to realize just how big a 24-person minimum is. Then we measure the outer circumference and the inner circumference of the 12 students seated in a circle. (You need a bunch of soft measuring tapes - we got ours as promotional items from a health care provider once, but you can also ask colleagues or students to bring them in.) You're better off doing this with enough measuring tapes to go around the circle - about 4-6 depending on their length.

After the students record the inner and outer circumference and then scale it up for 24 people, (with a small class you can use eight students and triple it,) we then continue in the Gemara, which explains that R. Yochanan is following Rebbi, and requires a Sukkah to be 4x4 amot. Then it introduces the idea that one person takes up 1 amah of space, and that for every 3 units in the circumference, there is 1 unit in the diameter. This is essentially approximating pi as 3. (If you're eager for fractions and decimals, don't worry - they're coming in just a few lines.) I don't recommend measuring a student to see if they take up 1 amah of space - that seems too likely to lead to bullying based on size. Instead, I always talk to the kids about how, as kids, they are smaller than adults, so we can expect our estimate to be lower than it would be if we were using adults, as Rabbi Yochanan was.

This activity is rife with opportunities for error, which is a great topic of conversation with the kids. (You can record them on the worksheet where it says "problem #2d.") For the inner circumference, even if you have each student hold the tape measure to his or her chest, you will find you have more of a polygon than a circle. (Around the outside there is less of an issue.) As I mentioned previously, middle schoolers are smaller than adults. What are your other sources of measurement error? Are the ends of the measuring tapes matching up properly? Is your measuring tape slanted, making it appear to be longer than it is? You cannot eliminate these sources of error, so be certain to point them out. It is important that students realize measurement is not simple or tidy.

Before we try to figure out the connection between a circle with a circumference of 24 amot and a square that is 4x4 amot, we convert the measurements to inches so the students can grasp them. I have the students use both the standard that an amah is 18 inches and the standard that an amah is 24 inches to convert. That way, we can compare our personal circle to the Rabbi Yochanan's. In most cases, your number will be on the low side. (Last year, for example, we got 430.4 inches, which is just below the range of 432-576 inches. After we have the conversation about measurement error, we then move on to the square and the circle.

To truly understand how this works, we need the Pythagorean Theorem. If we inscribe a square in a circle (which is what we are doing when we say the circular sukkah needs to enclose the minimum square sukkah), the diagonal of the square is also the diameter of the circle. To find the length of the diagonal of the square, we imagine that it is really two isosceles right triangles. Then we can figure out that the minimum side length of the square is 4 amot, so the diagonal is 4*root2. In fact, the Gemara says this, using 1 and 2/5 as its approximation of root 2. (This is actually quite close.) Then it "scales up" to figure out that 4*1.4 is 5.6 amot.  Essentially, we have concluded that the diagonal of the minimum sukkah is 5.6, so the diameter of the circle should be at least 5.6 amot. Before I return to the obvious Gemara issues, I note here that this part of the Gemara, the most difficult, is really well suited for older students - 9th or 10th graders who have studied Geometry could probably work independently through most of this, given a sheet written at their level. I've purposely over-explained some things to my 6th graders, but with a little less scaffolding, this activity could be beneficial to older students.

Now for the obvious Gemara issues. We just concluded that the minimum diameter should be 5.6 amot. Rabbi Yochanan wants to put a square with a diagonal of this length inside a circle which we concluded above actually has a minimum diameter of 8 amot (using the Gemara's pi.) This is a large discrepancy - an extra 2.4 amot bigger than the minimum diameter needed to encircle the minimum square. Rav Assi saves the day by clarifiying that 24 amot is around the outside, not counting the people inside the sukkah. Since the average person is 1 amah deep, he says (I leave this assertion unverified), we are dealing with a diameter of 6 amot - we cut off 1 amah on each end for the people who are standing there.

Rav Assi says we don't count the space that person A and person B on each end take up, leaving just a 6 amah diameter.

Now, if d is 6 amot, then c is 18 amot using the Gemara's calculations. The Gemara, familiar with fractions, notes that 5.6 * 3 = 16.8 and that even 18 amot is rounded up. They conclude that a lack of precision led Rabbi Yochanan to be stringent. I love this conclusion because at this age, I am often urging my students to be more precise. "Just about" and "almost" and "sort of" are the enemies of precision. (You'll note that this is a very different lesson than the one we did on pomegranates - math can have dialectics, too.) We said at the beginning of the lesson that is is incredibly difficult to be precise when measuring, so it's great to come back later to the importance of precision - at least when calculating. Taking pride in attention to detail is a value that is central to both Torah and Madda.