# Reconstructing a Circular Garden Wall

Let's assume for a moment that you have purchased some property and on this property you discover a beautiful garden and surrounding that garden you find a circular rock wall that has been partially demolished leaving only a portion of the wall standing. See the illustration below:

The portion of the rock wall that remains standing is the circular arc BC in the illustration above. If point A were known to you, it would be very easy to trace out the circular wall by simply attaching a string to this point and extending the other end of the string to some point on the circular arc and then rotating the string through 360 degrees of arc to complete the circle and hence the foundation for the remaining portion of the rock wall that is to be reconstructed. But, we don't know where the center of the circle is in our garden enclosure and thus point A is not given, but needs to be determined.

How would you as the landscape gardener go about determining where the exact center of the circular garden wall is so you could then trace out the remaining portion of the wall that needs to be rebuilt? I suppose you could take a guess at the center point of the garden wall, but chances are, you wouldn't get it right and so your wall would not wind up being perfectly circular. So, how can you find the center of the circular wall given only the remaining portion that is standing on your property; that is to say, wall BC represented by the arc of the circle shown above?

A method that can be used to determine the exact center of the circular garden wall is very easy to implement. All one needs to do is to mark off two chords (line segments along the arc whose endpoints lie on the arc) and which do not have to be of equal length, After marking off these two chords, if we take the perpendicular bisector of each chord (a ray passing through the mid-point of each chord at 90 degrees to the chord) and extend this ray out far enough so that both perpendicular bisectors cross each other, then they will cross each other at the center point of the circle. See the diagram below:

Now that we have determined the center point A of the circle which lies at the point where the two perpendicular bisecting lines cross inside the partially-demolished circular garden wall, we can now take our string from point A to a point on the wall (which will be the length of the radius of our circular garden wall) and rotate this string 360 degrees marking off the remaining portion of the circle from which we can then rebuild our circular rock garden enclosure. See the finished garden wall in the illustration below: