There’s nothing at all new about this, in fact it’s ancient, but maybe it’s new to you? So here goes.
Here’s a square inscribed in a unit circle:
Or to keep down the clutter, here’s just one quadrant of that:
One side of the square along with the radii at each endpoint forms a triangle. The length of one side of the square we’ll call and it is of course . That means the perimeter of the square is .
Now consider an inscribed octagon. Again, here’s just one quadrant.
One side of the square is labeled ; what’s its length? Well, drop a perpendicular from the vertex in the middle:
And now notice the right triangle with hypotenuse 1 and sides and is just half of a quadrant of an inscribed square, which means . Then . From that you can get and from that, . The perimeter of the octagon is then .
Well, that was so much fun, let’s do it again. Here’s a quadrant of a 16-gon:
The right triangle with hypotenuse 1 and sides and is just half of an eighth of an inscribed octagon, which means . Then , , and . The perimeter of the 16-gon is then .
You know the rest, right? From here you can do the 32-gon, 64-gon, 128-gon, and so on, getting , , , and perimeter . For , the perimeter is . As increases, the perimeter gets closer and closer to the circumference of the circle, so this gives a way to calculate .
What about circumscribed polygons? If you look at the figures above, you can see the ratio of the size of a circumscribed square to an inscribed square is . Likewise the ratio of the size of a circumscribed octagon to an inscribed octagon is , and so on. So the perimeter of a circumscribed -gon is . As increases, approaches from above. And the average of and is a better approximation to than either, though not by a lot.
This is something like the way Archimedes calculated around 250 BC (I told you this was ancient), although he started with a hexagon rather than a square and only went up to a 96-gon. He didn’t have Google Sheets, though.