I give a compass & straightedge construction using the Golden Mean which yields 3.14164 for pi at http://www.goldennumber.net/squaring-the-circle/ C.Ricci Ricci4.4428828Ricci4.4428828 (talk) 00:58, 11 June 2013 (UTC)
I have also discovered a compass & straightedge method of squaring the circle precisely in both area & perimeter on smooth Riemannian manifolds of positive Ricci curvature; despite the transcendental nature of pi: http://www.circleissquared.com/index.html C. Ricci Ricci4.4428828Ricci4.4428828 (talk) 00:58, 11 June 2013 (UTC)