What Tools Did the Greeks Use in Geometric Constructions?


Geometry had been used in agriculture, art and building long before Euclid, a prominent ancient Greek thinker and mathematician. Greek constructions, according to Euclid, formalized a practical mathematical operation into a field of academic study. Because he was attempting to prove mathematical theories in a logical and replicable way, Euclid standardized the tools allowed in problem solving. This included both actual tools and techniques for solving problems based on postulates that he proposed.


  • Geometric constructions are precise drawings of curves, lines and angles. "Pure" constructions, such as those studied by Euclid, depended only on relationships of lines and curves to each other. They used no numeric scale and had to be drawn with only a straightedge, compass and pencil.


  • Only two tools were allowed in Euclidian constructions: the straightedge for drawing straight lines and the compass for drawing accurate arcs. Use of these tools allowed accurate constructions that could be replicated by others. In addition to geometric concepts, the ancient Greeks were able to use these tools to construct "constructible numbers" which included all rational numbers and some irrational numbers and simple algebraic operations.


  • A straightedge was used to draw accurate lines. Often it was similar to a ruler without numbers. Alternative straightedges, such as a string lined with chalk, were acceptable to Euclid, provided they resulted in a consistently straight line.


  • The compass used a center point and radius measurement to draw accurate circles, curves and arcs. The Euclidian rules for geometry forbade markings on the compass and required that it collapse together when one arm was removed from the page. The compass could only be used to draw circles and arcs, according to Euclidian rules; it could not be used to measure distances.


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