A square with the side length of pi.

Question

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…

Answer 1

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.