how do you make a square out of a circle?

You're joking, right?

If cloth or paper, fold the circle into exact quarters. Then take a straight edge and draw a line across the bottom of the arc. You will have what looks like a triangle. Cut along the straight line you drew. This should be a square.

If the material is a solid which cannot be folded, again find the center point, then divide the circle into exact quarters. Draw a straight line between across each quadrant at the bottom of the arc to connect the sides of the square. Use a T square or L to get a true 90 degree angle. Then cut out your square.

You didn't mention any constraints (such as keeping the area or perimeter constant), so this approach is a transformation from circle to square and shows how the two shapes are related.

If you know the equation for the circle, you can modify the equattion so that it approaches the shape of a square.

For example, a unit circle has the equation x^2+y^2 = r^2 where r is the radius.

This is a specific form of equation x^n + y^n = r^n. If you make n as large as you like so that n remains an even integer, the shape of this curve will approach that of a square. Try it with Excel or any other simple program that plots shapes. As n increases by even integers, the shape morphs from a circle to a square.

Sort of a vague question, but you can place a square against the cirlce. Place it so the cirlce is on the inside of the square with the bottom of the circle touching the line of the square. Ensure the side of the circle is simulaneously touching the other edge of the square (perpendicular to the bottom). Draw a line along the side and bottom of the square. Flip the square over on the top of the cirlce so the cirle is again inside the square. Ensure the square's side is toucing the other side of the cirlce, and simultaneously the top of the circle is touching the inside line of the square. Draw a line along these two edges until you connect them with the previous lines. You know have a box enclosing a circle.

Draw 2 perpindicular lines through the center of the circle, dividing it into 4 quadrants. Then draw a straight line from the outside edge at one line to the next. That should get you a perfect square.