Park City Mathematics Institute
Geometrical Concepts from Constructions, Models, and Investigations

Circle Packing: A Directed Investigation of Descartes' Circle Theorem
Celeste Williams

Teacher Directions || Worksheet
    finalcirclepacking.doc - MS Word
    finalcirclepacking.pdf - PDF format
    circlepackingworksheet.doc - MS Word
    circlepackingworksheet.pdf - PDF format

Descartes' Circle Theorem involves relationships among radii of tangent circles, "packed" together. The relationship is expressed using the curvature of each circle, which is also the reciprocal of the radius involved.

The lesson is designed for high school geometry students who have studied tangent circles.

The lesson is designed to be done outside of class as a directed investigation.

No materials are required but the investigation can be done with the help of The Geometer's Sketchpad®.


  • Increase understanding of the curvature of a circle and its relation to creating tangent circles packed together.
  • See what can be learned from Descartes' Circle Theorem.
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    This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.