What are nice ways to draw a line of length $\pi$ if neusis is allowed?

Question

I understand that a straight line of length $\pi$ can't be drawn with a compass and straight edge without neusis.

I'm looking for a nice way to draw a line of length $\pi$ using compass and straight edge, where neusis is allowed.

I saw that you can draw a circle of radius 1, draw a line through its diameter, wrap a string around your circle, mark the string where it crosses the diameter line, and then straighten the string.

However, I'd like to avoid that kind of wrapping/unwrapping if possible, as the materials I'm working with are rigid, hypothetically.

Cheers!

Edit: It's become clear from the great comments that neusis doesn't get you the transcendentals. So I'd also be interested in the following:

1. Other techniques which do give the transcendentals, in particular $\pi$, like the rope stretching technique mentioned above.

2. Compass and straight edge processes, which when repeated, rapidly approach a length of $\pi$.

Actually I think the comments are enough to set me on the right path, but answers are still welcome. Thanks again!

Answer 1

What I am demonstrating is a construction that gives us pi, approximately, with an error of 0.0046%.

Draw a circle of a known radius, preferably a power of 2 and draw a vertical diameter. I am drawing the circle of radius 2 units.

Now, draw a perpendicular line at point B and cut 3 times of the diameter. I name that point to be F

Now, construct an angle $30^o$ at A as shown and complete the triangle. I am not showing the arcs.

Join F and G and divide the line segments by the diameter. In this case it is 4. So I will bisect the line twice.

That construction does not yield $\pi$. The result is $\sqrt{9+(\frac12+\sqrt{\frac34})^2}$ From this geometric solution you can get $3.141737211$ (closer to $\pi$) but not actually $\pi$. This is to say, you get $0.0046$ % error which is acceptable.

Answer 2

The following rational approximations to pi from Wolfram are good to 2, 4, 6 and 9 decimal places respectively. 22/7, 333/106, 355/113, 103993/33102.

I think this technique from YouTube for dividing a line using compass and straightedge will get you those indicated accuracies.

355/113 looks like the best effort to accuracy tradeoff. It gives 3.14159292035 versus 3.14159265359... for real pi. I think that's 0.99999991508 accuracy, or it overshoots pi by 0.0000002.66764189.

I still wish there was a way to apply an infinite series that approaches pi, using compass and straightedge to get arbitrarily close to pi with repeated applications of the procedure.

I think Nilakantha and Madhava's infinite series approximation to pi from this diagram on Wikipedia and YouTube video would work with the techniques mentioned above. Unfortunately this particular one takes 32 iterations to get the first 4 decimals of pi.

The formula is: 3 + 4/(2 x 3 x 4) - 4/(4 x 5 x 6) + 4/(6 x 7 x 8) - 4/(8 x 9 x 10) + ...