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I want to cut out a square with the side length pi. What is the most accurate method to do so?

P.S. you cannot use the already known fact that pi is approximately 3.14159…

Akshaya Gunnam
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    Draw a circle with radius $1$, wrap a thread around the upper semicircle, the length of this thread is then $\pi$. – QED Aug 26 '22 at 07:35
  • Or you could roll the piece of paper you want to cut into a cylinder with diameter 1 unit and then cut it. I don't know about accurate though. – Suzu Hirose Aug 26 '22 at 07:37
  • If you're allowed to use the continued fraction convergents $\ \frac{a_n}{b_n}\ $, you can always construct the $\ b_n^\text{th}\ $ part of a line of length $\ a_n\ $ with just a ruler and compass and a given line segment of length one. – lonza leggiera Aug 26 '22 at 09:43
  • $\pi$ in what units? Draw a square and define the length to be $\pi$. – user619894 Aug 26 '22 at 09:48

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Using ruler and compasses:

Draw a circle of radius 1 and draw a diameter across it. Draw a perpendicular to the diameter positioned at the centre of the circle, so you now have a circle with four quadrants. Then repeatedly bisect the angle in one quadrant. After, say, three bisections you will have an angle of $\frac{\pi}{2} \times \frac{1}{2^3} = \frac{\pi}{16}$. Set the compasses to the length of the small chord and along a new line mark off in succession sixteen such lengths. The total length then marked along the line will be approximately $\pi$ (actually 3.137 rather than 3.142). Use more bisections to get a better approximation.

WA Don
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