Using just a straight edge and compasses, it has been proven that it is impossible to construct a square with the same area as a given circle.

What confounds me, even now, is that Archimedes, introduced his eponymous spiral into the mix and came up with a solution.

I mentioned it in passing in https://www.talkgraphics.com/showthr...555#post619555 that I was looking a bit more deeply into what he got up to with his spiral: one turn inscribes one-third the area of its enclosing circle. My Area of Archimedean Spiral.xar tried to depict this in a XAR file to see how accurate Xara's implementation is. I got 99.843% accuracy.

My final diagram tried to follow Archimedes' construction logic, using XDPX, and I got Area of Circle.xar, where the green circle, the yellow triangle and the blue square should all have the same area (three times the red sleeping bird).
The method involves drawing the tangent to the spiral at the end of the first turn. This bit is a bit ropey.

In the end I managed 99.852%.

So Xara, you could do better!

Acorn