
Finding the center of a circle or arc
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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 centerfinding 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
NonEuclidean constructions
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