A circle of unit radius is completely covered by three identical squares. What is the smallest size of the squares?

(In reply to re(2): overlap solution by Charlie)

This measurement is confirmed in Geometer's Sketchpad by the following construction:

Construct an equilateral triangle and find its center via two intersecting angle bisectors. Construct a line segment from the center to one of the vertices.  Conscruct a circle centered on the center of the triangle with this new line segment as a radius.

Then construct a perpendicular to the side opposite the vertex to which the circle radius had been drawn, through an endpoint of that side (which is a chord of the circle); call that endpoint E.  Bisect the right angle that's in the quadrant of that 4-way right angle that includes the center of the circle.  Where that bisector intersects the previously drawn radius, is point B, which will be one corner of a square--the only corner within the circle. Construct a line through the center of the circle, C, parallel to BE, and construct a tangent (perpendicular to this new parallel line) where it intersects the circle.  Construct a ray from point B through point E, and where it intersects that perpendicular (tangent line) is an external corner of the square, and the measure of its distance to B is one side of the square (be sure to calculate this divided by the length of the circle radius to reduce the scale to one where the radius is 1).

The resulting value in Geometer's Sketchpad is 1.25882, confirming the value found before.

Posted by Charlie on 2005-10-21 22:45:26