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Finding the center of a circle or arc
Geometry construction using a compass and straightedge

This page shows how to find the center of a circle or arc with compass and straightedge or ruler. This method relies on the fact that, for any chord of a circle, the perpendicular bisector of the chord always passes through the center of the circle. By applying this twice to two different chords, the center is established where the two bisectors intersect. A Euclidean construction

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

The image below is the final drawing from the above animation.

  Argument Reason
1 RS is a chord of the circle C A chord is a line segment linking two points on a circle. See Chord definition.
2 AC is the perpendicular bisector of the chord RS AC was drawn by constructing the perpendicular bisector of RS. See Constructing the perpendicular bisector of a segment for the method and proof
3 BC is the perpendicular bisector of the chord PQ As in (1) and (2)
4 The center of the circle lies on the line AC. The perpendicular bisector of a chord passes through the center of the circle. See Chord definition.
5 The center of the circle lies on the line BC. As in (4).
6 Point C is the center of the circle. The only point common to both AC and BC.

  - Q.E.D

Try it yourself

Click here for a printable worksheet containing two center-finding problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions