I can't speak to the Pythagorean Theorom, but I can say that the Scioto Hopewell (most likely knew about pi and now to use it to calculate the area and circumference of circles, based on the geometry involved in their most famous site.

The site in question is the Newark Earthworks in Ohio, which were constructed around 250 CE. As you can see on the survey image, the complex includes several circular features. The two largest are known as the Observatory Circle (upper left) and the Great Circle (lower center). To give you a better sense of the scale, this is what The Observatory and the Great Circle look like today. Zoom in and notice how tiny cars and houses are in comparison.

The diameter of the Observatory Circle is approximately 1050 feet (1054 to be more precise), appears to be a common unit of measure at several other Ohio Hopewell sites. This image shows the regularity between five prominent Ohio Hopewell sites, though it does have a typo (saying 1500 instead of 1050). Among archaeologists the “Observatory Circle Diameter” (OCD) has been recognized as the standard unit of measurement at Scioto Hopewell sites (though in reality the Hopewell’s standard unit was probably some fraction of this- given the construction methods involved we should probably talk about the Observatory Circle Radius instead). Not only does it show up in the diameter of most major circles, it also shows up in the spacing of features at sites. For example, the center of the Observatory Circle is 6 OCDs from the center of the Great Circle while the center of the Observatory’s Octagon is likewise 6 OCD away from the Square.

What makes the Newark Earthworks interesting with regards to pi is the relationship between the sizes of the Observatory Circle, the Great Circle, and the Square (more properly known as the Wright Earthwork). The areas of the Observatory Circle and the Square are the same. Likewise, the Square's perimeter is the same as the Great Circle's circumference. There is a very slight error in the Great Circle's construction that allows us to know that it was constructed in two large arcs.

The implications here are that the Ohio Hopewell were able to do the geometric calculations to produce squares from circles, circles from squares, and determine the areas and circumference / perimeters of both. We use pi to do these calculations today, but we're not sure whether the Ohio Hopewell used pi, tau, or perhaps some other unknown method to construct this complex, but they were definitely doing some related geometry and measurements in the process.

Now, I’ve been calling the circle-and-octagon complex “the Observatory,” but I haven’t yet explained why. It certainly doesn’t look much like observatories you’d likely be familiar with. Back in the 1980s, when solar alignments were first detected at Stonehenge, Ray Hively and Robert Horn were teaching a course on ancient cosmology at Earlham College in Indiana. Upon hearing the news from Stonehenge they were skeptical. Any sufficiently complex structure, they hypothesized, would have solar alignments by pure chance. To test this hypothesis, they took their class on the round and went to Newark, Ohio, hoping to find solar alignments at the Circle and Octagon. To their surprise, they didn’t find any. On the way home, a student asked: “What about lunar alignments?” On the next trip to Newark, things started to fall into place.

The major axis of the Observatory complex aligns with the northernmost rising lunar standstill, part of an 18.6-year long cycle of moonrises and moonsets. A standstill is the point in the cycle where the moon appears to stop and reverse direction. The rising and setting of the moon bobs north-and-south over the course of the cycle, so there are eight major standstills - northern maximums and minimum for rising and setting, southern maximums and minimums for rising and setting. Minor alignments for the other seven were eventually found for the Observatory as well, though whether all were intentional like the major axis alignment or whether they were side effect of that the major alignment is something that’s still debated (though more and more people, including Hively and Horn, and rejecting mere chance).

If you refer back to the map of the entire site, you’ll notice a pair of parallel embankments heading south from the Observatory Circle. This is the northern terminus of the Great Hopewell Road. From Newark, it heads off in the direction of Chillicothe, Ohio, 60 miles away. So far the southern terminus hasn’t been found, but parts of the Great Hopewell Road continue to be found along the way, some we might still find out where exactly in Chillicothe it ends. Chillicothe is important because it was the epicenter of the Scioto Hopewell culture, and had many earthwork complexes in a fairly compact area. Referring back to the five sites that share similar consistent sizes for their component features, you can see all of them on this map of the Chillicothe area. Of interest here is the High Banks Earthworks, which are a bit hard to see on that map but they’re circled in yellow in the southeast. Here you can see them better, both in the context of the immediate Chillicothe area and with a map of the site itself. You’ll of course notice that we once again have a circle-and-octagon combination.

So do the High Banks Earthworks also have lunar alignments? Maybe. While the site is currently protected, that protection came too late to preserve its Circle-and-Octagon. Studies have been done using the survey map, in which case the alignments don’t fit as well as the ones at Newark. The major axis is close to aligning with the southernmost rising standstill, but if the old map is accurate, it’s off by a few degrees as illustrated here. Did the surveyor mess up? Did the Scioto Hopewell? Did the Hopewell even intend to align the major axis of this site with a lunar standstill? Questions that will have be answered another day. Now that Modern Lidar let’s us see the remains of the old structure again, someone will hopefully get around to double-checking Hively and Horn’s old analysis based on the survey data.