Constructing the Centre of a Circle

Compass and straight edge constructions are of interest to mathematicians, not only in the field of geometry, but also in algebra. For thousands of years, beginning with the Ancient Babylonians, mathematicians were interested in the problem of "squaring the circle" (drawing a square with the same area as a circle) using a straight edge and compass. This problem is equivalent to finding the area of a circle. It turns out that this is impossible, but no-one managed to prove this until 1882!

However, it is possible to construct the centre of a given circle, using only a straight edge and compass. For this construction, you will need a straight edge (ruler - but you won't be measuring anything), a pair of compasses, a pencil and paper. I have drawn the pictures using the robocompass app. It's fun to play with, and you can use it to do all sorts of geometric constructions. There's a little bit of coding to learn, but a list of instructions is provided. Once you've written your little program, you can invite a few friends over, and relax while you watch the construction. Be warned that, when you are using it to draw arcs, the robocompass compass point may appear to be a little away from the centre, but the drawing is actually accurate.

Note: these instructions assume that you are familiar with constructing the perpendicular bisector of a straight line. We are going to do that twice. If your memories of this construction are a bit hazy, it's probably a good idea to re-read the article on finding the perpendicular bisector of a line segment before trying to do this construction.

The Construction

Step 1: Start with any old circle like the one shown in the picture.

Step 2: Draw two chords inside your circle, wherever you like. It's best to make sure they aren't parallel. It's even better to use a ruler to draw them. The picture on the right shows the two chords that I drew, but any pair of chords that isn't parallel is fine.

Step 3: Construct the perpendicular bisector of one of the chords. Make it nice and long.

Step 4: Construct the perpendicular bisector of the other chord. Make it long enough to intersect the other perpendicular bisector.

Step 5: The point where the two perpendicular bisectors drawn in steps 3 and 4 intersect is the centre of the circle. Label it with an \(O\), or whatever you want to call the centre of the circle. I've always liked the name Steve.